One voltage added to two readings

The millivolts in the wire ends with a list of things it does not claim, and one of them is that subtracting the two ends of the shared conductor removes the error. It removes most of it, and the amount left is a measurement: the rejection four resistors decide takes exactly that circuit and finds a difference amplifier built from tenth-per-cent resistors removing 53.99 decibels of it, against a closed form’s 53.98 — a factor of five hundred, set by one plus the gain over four times the resistor tolerance, with the amplifier having nothing whatever to do with it.

Five hundred is a good number and it is a tolerance. There is a second arrangement that removes the same error, and what limits it is not a tolerance at all.

Why the difference is clean #

The interfering voltage is one voltage. The load’s current crosses the shared conductor and lifts the local node above the star point by a definite amount, and both sensors are referenced to that node — so both readings are high by exactly the same number of microvolts.

A difference between two such readings therefore has none of it. Not a small amount of it, and not an amount set by how well two components match: none, because the quantity being subtracted is literally the same quantity in both terms. With the two sensors’ own stubs equal, the residual measured at a kilohertz is 2.5 picovolts against 511 microvolts in either reading — 166 decibels, and the 2.5 picovolts is itself identifiable rather than being the solver’s noise. It is the two sensors’ input dividers drawing different currents, 5 nanoamps against 2.5, through a milliohm of stub.

The difference from the difference amplifier is worth stating sharply, because both arrangements are described as “measuring differentially” and they are not the same idea.

A difference amplifier subtracts two voltages that are not equal. The interference appears at both of its inputs and the amplifier’s job is to reject what they have in common. How well it does that is decided by how nearly its four resistors form two equal ratios — nothing about the amplifier enters — so the answer is a component tolerance and no arrangement of good parts makes it exact.

Two sensors on one node subtract two readings that contain the same term. There is no rejecting to be done: the term cancels in the arithmetic, before any amplifier sees anything. What is left is whatever the two readings do not have in common, which is a property of where the conductors meet.

What is left, and it is a length rather than a tolerance #

So the residual is the asymmetry, and locating it took a wrong answer first.

The obvious guess is that the residual is proportional to how much copper each sensor has to itself — the stub between its own reference and the shared node — because that is the part of the return that is not shared. Measured, it is not: the residual is 1.71 nanovolts at half a millimetre of stub and 1.72 at ten millimetres, flat across a factor of twenty.

The reason is that a stub matters only if there is a current in it, and in that first arrangement there was not. A sensor read by a high-impedance input draws nanoamps, so its stub carries nanoamps, and a milliohm carrying five nanoamps is 25 picovolts whatever its length. The stub’s length cannot matter because nothing is flowing through it.

What makes a stub matter is a real current, and a sensor has one: its own supply current, which comes in from somewhere and goes home through the same stub. Give each sensor a milliamp of its own and the arithmetic becomes:

$$v_\mathrm{residual} = I_\mathrm{sensor}\,(Z_2 - Z_1)$$

— the sensors’ own current times the difference between their two stubs. Two stubs of equal length give zero. Two stubs differing by a stated fraction give that fraction of one stub’s impedance:

Exactly ten nanovolts per per cent, against the closed form’s ten. The arrangement’s quality is how nearly two pieces of copper are the same length, times how much current the sensors draw, and neither of those is a component tolerance. Both are things a layout controls directly, and one of them — the current — can be made small by a choice that costs nothing.

That is a better kind of number to be limited by, and the comparison says how much better. A difference amplifier’s 54 decibels needs tenth-per-cent resistors and cannot be improved by care in the layout. This arrangement’s 74 decibels needs two stubs matched to ten per cent, which is a routing decision rather than a purchase, and matching them to one per cent — which is easy, since they are drawn together — gives 94.

The rejection has no corner in it #

The number that decides whether an arrangement is worth using is rarely the one at direct current, and this is where the two arrangements part company by a factor of tens of thousands.

Measured across six decades, this one’s rejection is 74.17 decibels at every frequency — 100 hertz, 10 kilohertz, a megahertz, ten megahertz — varying by less than a twentieth of a decibel over the whole sweep. Both quantities in the ratio are drops across the same copper. Each reading’s error is the load current through the shared run, which is resistive and then inductive; the residual is the sensors’ own current through the difference between two stubs, which is resistive and then inductive in exactly the same proportion because both stubs are the same kind of track. The corner at 79.6 kilohertz is in the numerator and the denominator alike, and it cancels.

A difference amplifier’s does not. The corner the instrument has no part in finds a 95-decibel instrumentation amplifier falling twenty decibels a decade above 290 hertz when its source has a kilohm of imbalance and ten picofarads at each input — the mechanism being a difference of two time constants rather than anything about the part, which is why the essay is titled as it is. Put the two side by side at the frequencies that matter:

The amplifier wins below about three kilohertz and loses everywhere above it, and by a megahertz the difference is fifty decibels — a factor of three hundred in the residual. Which is the practical reading of the whole essay, because the interference this field is about is not at direct current: the previous essay measures a switching load putting a flat comb of 0.828 millivolts a tooth into a shared reading from a hundred kilohertz upwards, and every tooth the same height is entirely in the band where the amplifier has stopped rejecting and this arrangement has not started to.

The reason for the difference is structural rather than a matter of degree. A difference amplifier rejects by balancing two paths, and two paths balance only as well as their time constants match, so there is always a frequency above which they do not. This arrangement rejects by arithmetic on two numbers that contain the same term, and arithmetic has no frequency in it.

Where the arrangement is already used, without being described this way #

Three arrangements in ordinary use are this one, and in each the reason given for it is something else.

A ratiometric bridge measurement. A strain gauge bridge excited from a supply and read against the same supply as its reference has the supply’s noise in both the reading and the reference, so it divides out. The usual account of that is about supply noise, and it is the same cancellation: the two leads nobody counts finds the excitation leads scaling every reading by exactly $-2r/(R + 2r)$, and a ratiometric reading removes that term for the same reason a difference of two sensors removes this one — it appears identically in two places and the arithmetic subtracts it.

A reference junction in a thermocouple measurement. Two junctions at different temperatures in series, with the cold one at a known temperature, is a difference of two readings that share a return. Everything common to the pair goes, which is why a thermocouple can be read to microvolts on a board carrying amperes.

And a quadrature pair in a resolver or an encoder. Two sensors on one shaft, read against one reference, with the angle computed from the ratio. The ratio removes every term common to both — the excitation’s amplitude, the temperature coefficient of the coupling, and the shared return’s drop — and the residual is the mismatch between the two channels, which is exactly the quantity this figure sweeps. The rejection the parts have is the ceiling the same mismatch puts on an instrumentation amplifier, and the arithmetic is the same in both places: a matched pair of mediocre channels costs nothing and a mismatched good pair sets a limit.

The common shape is worth naming because it is a design rule rather than three coincidences. When an error appears identically in two readings, take their difference or their ratio and the error is gone exactly; when it appears differently, reject it and the answer is a tolerance. The first is available whenever a measurement can be arranged to need two readings, which is more often than it is used, and the arrangement is decided by the routing rather than by the parts.

What the sensors’ own currents do on the way home #

There is a term in the numbers above that the account so far has passed over, and it is worth pulling out because it is the one place the two arrangements are alike.

Each sensor draws a milliamp, and that milliamp does not stop at the local node. It crosses the shared run too, on its way to the star point. So the shared conductor carries a hundred milliamps of somebody else’s load plus two milliamps of the sensors’ own, and the local node is lifted by both: 500 microvolts from the load and 10 from the sensors, which is the 511 microvolts either reading is wrong by rather than the 500 the essay before it would predict.

Two things follow and they point in opposite directions.

The sensors’ own contribution cancels too. It is a common term like any other — one voltage on one node — so the difference between the two readings has none of it, and the eleven microvolts is not a residual. That is worth noticing because it is not obvious: a sensor corrupting its own reading through a shared conductor sounds like a self-inflicted error that no differencing could remove, and it is removed for exactly the reason the load’s is.

But it is the same current that sets the residual. The milliamp that cancels in the shared run is the milliamp that does not cancel in the mismatched stubs, so the arrangement’s accuracy and the size of its own self-interference are set by one quantity. Halving the sensors’ supply current halves the residual and halves the eleven microvolts, and both are improvements; there is no exchange here, which is unusual on this site and is worth saying plainly.

It also names the design choice. A sensor whose supply current returns somewhere other than through its own reference stub — a separate supply return, or a sensor powered from the measurement side — takes the current out of the residual entirely, leaving only the picovolts of input-divider asymmetry. That is the arrangement a four-terminal measurement is, applied to a supply rather than to a sense: two terminals measure the leads as well is the same separation of a current path from a sense path, and the same reason it works.

Neither reading is any better than it was #

One thing this arrangement emphatically does not do, and it is worth being blunt because the figure’s two curves invite the wrong reading.

Each individual reading is still 511 microvolts wrong. Nothing about putting a second sensor on the node improved the first one. If the measurement needs an absolute value — this temperature, not the difference between two temperatures — the arrangement offers nothing at all, and the remedies are the ones the essay below lists: a shorter shared run, its own return to the star point, or a difference amplifier and its 54 decibels. The first of those is less available than it sounds — where the current comes back shows a return choosing its own path above about a hundred kilohertz whatever the layout intended.

So the arrangement is a way of choosing what to measure, not a way of measuring better. That is a genuine restriction and it is also where the leverage is: a great many measurements that are taken as absolute are wanted as differences, and the cost of arranging for two sensors instead of one is usually less than the cost of the copper that would make one sensor accurate.

The distinction also decides what the failure looks like when the arrangement is wrong. A shared-impedance error in an absolute reading moves with somebody else’s load and looks like drift. In a difference it does not appear at all — until the two stubs are cut to different lengths at some later revision, at which point a quantity that was exact becomes a hundred nanovolts, with nothing in the schematic having changed. The schematic cannot hold this arrangement’s accuracy, because what it depends on is two lengths, and a schematic has none.

Two routes, and what the residual tells about them #

The site’s habit is to compute a quantity twice by routes sharing nothing but the netlist, and this figure has an unusually informative pair.

The first route is the whole thing solved at once: five elements plus two sensors plus two input dividers, solved at a frequency, the two node voltages read, and the difference taken against what the difference ought to be. The second is the closed form — the sensors’ own current times the difference between the two stubs’ impedances — which contains no load, no shared conductor, and no input divider.

They agree to five per cent, and the residue is the thing the essay is about arriving in its own verification. The closed form has no input dividers in it, so it is missing the 2.5 picovolts that the matched case measures, and it has no shared conductor in it, so it is missing the small asymmetry the sensors’ own currents produce by crossing the shared run on their way home. At a ten per cent mismatch those are parts in ten thousand of a hundred nanovolts, which is why five per cent is a comfortable bound.

One detail in the first route had to be repaired and it is the kind that does not announce itself. The errors must be complex differences and not differences of magnitudes. Written the second way, the matched case came out at 1.78 nanovolts rather than at picovolts, because a reading whose interfering term is partly reactive has a magnitude that is not the sum of the two magnitudes — so subtracting magnitudes charges the arrangement with a quadrature term no instrument would report. Two of the three quantities here are differences of numbers that nearly cancel, which is exactly where that matters, and the wrong version looked like a real limit on the arrangement rather than like an arithmetic slip.

Still open: the supply current that is not a constant, and the third sensor #

A sensor current that varies. The residual is the sensors’ own current times a length difference, and this figure takes that current as a constant milliamp. A real sensor’s supply current moves — with temperature, with its own output, and with whatever it is doing — so the residual is not a fixed offset but a term that tracks the sensors’ activity. Sweeping it would say whether the arrangement’s error is a calibratable offset or a signal-dependent one, and those are very different things to live with.

Three sensors, and which differences are clean. With two sensors there is one difference and it is exact. With three there are three differences and they cannot all be independent, so the arrangement’s error structure is a matrix rather than a number, which is the shape what a network answers sets up for a solve with several sources in it — and whether a triple of readings can be combined so that every pairwise difference is clean, or whether one of them has to carry the residual, is a question about the topology that this figure’s two-sensor case cannot answer.

The comb, differenced. Every tooth the same height measures what a switching load puts into one reading, and this essay measures what two readings can do about a sinusoid. The two together are the useful case and neither draws it: a flat comb rejected by a flat seventy-four decibels leaves a flat comb seventy-four decibels down, which would be the first repair in this field whose residual has no corner in it at all — and worth drawing for exactly that reason.

And the same cancellation at the excitation. Everything here is about the return. Two sensors excited from one supply through one shared conductor have a common term in their excitation rather than in their reference, which divides out of a ratio rather than subtracting out of a difference — so the arrangement’s quality is then a matching of two gains rather than of two lengths, and the two mechanisms are not interchangeable.

What is checked #

The rejection with the stubs matched is required to be exact rather than good — the residual below a millionth of either reading’s error — because that is the whole claim that separates this arrangement from a difference amplifier’s 54 decibels.

The residual is required to be proportional to the mismatch, across four settings from one per cent to thirty, to five per cent of proportionality, and against its closed form. Either alone would be weaker: the proportionality without the closed form would leave the constant unmeasured, and the closed form at one setting would not say it is a proportionality.

The errors are complex differences. Stated as a requirement rather than as a comment because it is a fact about the measurement: subtracting magnitudes reported 1.78 nanovolts where the answer is 2.5 picovolts, which is a factor of seven hundred, and it was in the direction that makes the arrangement look worse rather than better — a failure that would have survived review.

And each individual reading is required to carry the error that the difference does not, by a factor of at least twenty at every frequency in the sweep. A figure that showed only the clean curve would be a figure about a circuit with no problem in it.