The two leads nobody counts

The leads that are in the bridge found a strain gauge on the end of two long wires being indistinguishable from a strain gauge under load, and fixed it by moving one of those wires into the arm beside the changing one. The error went from 999 microstrain that is not there to exactly zero, and a twenty-kelvin drift from 76.8 microstrain to 0.077.

A bridge has more than two wires. That is the same Kelvin argument two terminals measure the leads as well makes about a four-terminal resistance measurement, applied to a bridge. A quarter bridge with three-wire sensing has three wires going to the gauge, and it also has two going the other way — from the instrument’s excitation supply to the bridge’s two excitation terminals — and those two carry every milliamp the bridge draws.

They do not produce an offset and they do not produce a nonlinearity. They produce a gain error, which is the one class of fault the three-wire fix cannot touch, the one more arms do not cancel, and the one that a calibration removes until something moves.

A divider nobody drew #

The arithmetic is a two-resistor divider and it is worth writing down before anything is solved, because the closed form is what the solve is checked against.

A 350-ohm bridge presents 350 ohms between its excitation terminals — two 700-ohm legs in parallel — and the two excitation leads are in series with it, one at each end. So the bridge is excited at R/(R + 2r) of the supply’s voltage, and every reading is scaled by exactly that.

With 0.35 ohms in each lead the excitation is 9.98004 volts of a nominal ten, and the scale factor is low by 1,996 parts per million. Solved on the netlist rather than computed, the reported strain at a thousand microstrain is low by 1,995 — the difference is the quarter bridge’s own load on its excitation moving very slightly with the strain, which is a part in a thousand of the error and not of the reading.

Two thousand parts per million is a large number in this field, and it is a great deal more than the voltmeter four wires do not remove finds left after a Kelvin connection has done its work. It is a fifth of a per cent, which is beyond most instruments’ gain specification and well beyond the tolerance of the gauge factor a user types in. And it is entirely invisible: the reading is linear, repeatable and stable, and there is nothing about it that looks wrong.

Only the sum of the two leads matters, which is worth knowing because it means the pair need not match. Half an ohm on one side and two tenths on the other gives 997.008 microstrain, and so does 0.35 on each, and so does the whole 0.7 in one lead — identical to three decimal places, because the two are in series in the same loop.

More arms do not help, and that is the surprise #

The bridge that is linear near one point found the quarter bridge’s nonlinearity — the output is V·δ/(2(2 + δ)) rather than V·δ/4, low by half the fractional change — and found that a half bridge with its pair in opposite directions cancels it and a full bridge cancels it exactly.

That fix does nothing at all here. At a thousand microstrain the nonlinearity falls from 999 parts per million on a quarter bridge to zero on a half and zero on a full one; the excitation leads’ gain error is 1,995, 1,996 and 1,996 parts per million for the same three. It cannot cancel, because it is not a property of how the arms are arranged — it is a property of what is in series with all four of them at once.

So a full bridge read four wires deep has no nonlinearity and a two-thousand-part-per-million gain error, and a designer who has moved to a full bridge to get rid of the first fault has left the second, larger one exactly where it was. The two faults are of comparable size on a quarter bridge at full scale, which is probably why they have never been separated: fixing one and measuring again shows an improvement, and the improvement is real, and the remaining error is the other one.

A gain error across the whole span #

The error is the same fraction of the reading at every strain, and that is the difference between it and everything else measured about this bridge.

Across a span from zero to two thousand microstrain the excitation leads contribute −1,995.0 parts per million at every point, spreading by 1.8 parts per million over the whole range. That residual spread is the quarter bridge’s own load on the excitation changing with δ; on a full bridge it is zero, because a full bridge’s resistance between the excitation terminals does not depend on the strain at all.

A pure gain error has one very convenient property: a two-point calibration removes it exactly. Load the structure with a known force, read the instrument, scale. Nothing is left, at any strain. That is why this fault has not caused more visible trouble than it has — every calibrated system has already absorbed it.

And it has the matching inconvenient property: the calibration is a calibration of the cable. Change the cable, shorten it, splice it, warm it, and the gain error moves and the calibration does not know. The arm-lead error is worse in a different way — it is a nonlinearity as well as a span error, so no two-point calibration removes it — but it is at least attached to the gauge, which is where a user expects the sensitivity to live.

What happens when the cable warms #

Copper’s temperature coefficient is the same number a winding drifts at, and it is taken from the magnetics field’s own dissipation model rather than typed a second time here.

With 0.35 ohms in each excitation lead and 0.35 in the arm leads with three-wire sensing in place, warming every lead by twenty kelvin changes the reported strain by 0.2296 microstrain on a thousand-microstrain reading. Six wires bring that to 0.07676, and eighty kelvin gives 0.9177 and 0.3070.

The interesting comparison is not with zero. It is with what three-wire sensing left behind. The arm leads alone, with three-wire sensing, drift by 0.0768 microstrain over twenty kelvin — that essay’s own residual, and the number it was pleased with. The excitation leads drift by 0.153 over the same twenty kelvin, which is twice as much.

So the three-wire fix, applied on its own, leaves the larger of the two lead errors in place. Sweeping the excitation lead’s resistance against that residual shows where the crossover is, and it is not far: even 35 milliohms of excitation lead — a few centimetres of heavy wire — contributes 0.092 microstrain against the arm leads’ 0.077, so on any cable at all the excitation pair dominates.

Two more wires, and what they are worth #

The fix is to stop assuming the excitation and measure it. Two more leads from the bridge’s own excitation terminals back to the instrument, which ratios the output against what they report rather than against the supply.

Those two leads carry only the instrument’s input current — the quantity the current the instrument draws is about — so their own resistance is divided by the instrument’s input impedance before it reaches anything. At 0.35 ohms of sense lead into a megohm the residual is 0.35 parts per million; into ten megohms, 0.035; into a hundred kilohms, 3.5. It is the lead over the input impedance, exactly — a ratio of two resistances with no bridge in it — to a tenth of a per cent across five decades.

That puts a specification on the instrument that is easy to meet and easy to get wrong. A megohm is enough to put the sense residual three decades below anything else in this essay. Ten kilohms is not: 35 parts per million is a larger error than most of the things a strain instrument is specified on. And it is the one number in the whole arrangement that has nothing to do with the cable.

The second, more practical thing six-wire sensing gets wrong is where the wires are taken from. It is worth exactly as much of the excitation lead as it reaches past: tapped at the instrument’s own terminals it removes nothing, tapped at the gauge it removes everything, and tapped at a connector halfway down it removes half. Half of 1,995 parts per million is 998, which is still a large number — and a connector halfway down is where the sense wires usually go, because that is where there is room.

The other fix, which needs no extra wires #

Exciting the bridge from a current source instead of a voltage source removes the same error and needs four wires rather than six.

The reason is the one the bridge that is linear near one point already gave for a different purpose: a current source sets the current through the bridge whatever is in series with it, so the lead resistance changes the voltage at the excitation terminals and not the current through the arms. The reported strain is then I·R·δ/4 over I·R, and the leads are not in it. Measured, the wiring error is 4 × 10⁻⁴ parts per million at 0.35 ohms of lead and 0.01 at ten ohms, which is the solver’s own arithmetic rather than a residual.

Current excitation also halves the arm-lead drift — 0.0385 microstrain over twenty kelvin against 0.0768 — and it halves the nonlinearity, which is that first essay’s result. Three improvements from one change of source, and the first essay already recommended it for the third.

A current source has difficulties of its own as an instrument, which the ammeter that is a resistor sets out from the other direction. What it costs here is compliance and heat. The source has to develop the bridge’s voltage plus both lead drops, so a long cable eats headroom that a voltage source would not have needed; and the power in the bridge is set by the current rather than by the supply, so a resistance that drifts changes the dissipation. Neither is fatal, and neither is discussed as often as the six-wire connector.

How large it is in the units somebody reads #

Microstrain is the wrong unit for deciding whether this matters, because nobody buys a strain gauge to measure strain. Putting the same number into the two applications that use bridges settles it.

A load cell is a full bridge on a machined element, sold with a full-scale output in millivolts per volt of excitation and a calibration certificate. A two-thousand-part-per-million gain error is 0.2 per cent of reading — ten to twenty times the non-linearity a good cell is specified at, and comparable with its whole accuracy class. The certificate hides it, because the cell is calibrated with its own cable attached; and a user who shortens the cable to fit the machine has invalidated the certificate by more than every other term in it combined.

A strain measurement on a structure is usually uncalibrated: a gauge is bonded on, a gauge factor is typed in, and the reading is believed. There the 1,995 parts per million is a straight 0.2 per cent error on every number, and it sits underneath a gauge factor whose own tolerance is often quoted at one per cent. That is the honest reason it has never been a scandal — it is smaller than the largest term that is already accepted, and it points the same way every time.

What changes that judgement is a comparison between two channels. A twelve-channel instrument with cables of different lengths has a different gain error on each, and the differences between channels are what a strain survey is usually about. Two channels whose cables differ by a metre of 0.1-ohm-a-metre wire differ by about 570 parts per million of reading, and nothing in the data says so. It is the same failure of a shared conductor that the millivolts in the wire measures on one return, moved into the excitation.

Finding it without adding wires #

There is a measurement that exposes the fault with four wires, and it is worth stating because it is also how the model is checked against a real cable.

Excite the bridge, measure the output at a known load, then add a known resistance in series with one excitation lead and measure again. The reading changes by −2Δr/(R + 2Δr + 2r) of itself, which is a single number with the bridge’s own resistance in it, and solving it backwards gives the lead resistance that was already there. An extra ohm in one lead on a 350-ohm bridge moves the reading by about 5,670 parts per million, which is easy to measure and hard to mistake for anything else.

The same measurement says whether the instrument is doing its own sensing internally, which several do without documenting it. If adding the resistance moves the reading by the full amount, the excitation is assumed; if it moves it by nothing, the excitation is sensed somewhere the added resistor is inside the sense loop. That distinction cannot be read off a specification sheet and is the whole of the difference between a 0.2 per cent error and none.

Four faults, and the wiring that removes each #

The three measurements now form a list of four faults and the wiring that removes each, and it is worth setting them beside each other because no two are fixed by the same thing.

The quarter bridge’s nonlinearity is half the fractional change, and it is removed by more arms or halved by current excitation. The arm leads’ span error and drift are removed by moving a lead into the neighbouring arm, which is three-wire sensing. The excitation leads’ gain error is removed by sensing the excitation at the bridge or by exciting with a current, which is the kind of arrangement whose rejection the corner the instrument has no part in depends on. And the sense leads’ own residual is the ratio of a lead to an input impedance, removed by an instrument that does not load them.

Two of those are about where the wires go and two are about what the source is, and the pair a designer is most likely to have got right is the first pair. The gain error is the one with no symptom: it is linear, repeatable, stable at constant temperature, and calibrated out by the first person who ever loads the structure.

Still open: the gauge’s own self-heating, the cable as a capacitance, and the current source’s noise #

The dissipation the leads move. The excitation leads drop volts and therefore burn watts, and they take those watts away from the bridge — 0.2 per cent of the power at 0.35 ohms a lead. Under current excitation they take none away, because the bridge’s current is fixed, so the gauge runs hotter with a current source than with a voltage source of nominally the same excitation. A gauge that runs hotter reads differently, through the gauge’s own temperature coefficient and through whatever it is bonded to, so the wiring choice has a thermal consequence as well as an electrical one. Sizing it needs a thermal resistance nothing solved here carries.

The cable as something other than a resistance. Everything solved here is direct current, and a cable has capacitance between its conductors and to its screen. On a bridge read at direct current that is harmless; on one read with an alternating excitation — which is what a carrier-frequency amplifier does, and what removes the instrument’s own offset drift — the same capacitance forms a divider with the gauge’s resistance and puts a phase shift between the output and the reference. The question that measurement would answer is what excitation frequency a cable of a given length allows before the capacitance costs more than the offset drift it was chosen to remove.

Whether the current source is quiet enough. Current excitation removes three errors and replaces the voltage source with a current source, whose own noise multiplies the bridge’s output directly. A voltage-excited bridge read ratiometrically is immune to its supply’s noise by construction — the same noise appears in the reading and in the reference and divides out — and a current-excited one is immune only if the current is the reference too. Whether that arrangement is available, and what it costs in the current source’s own stability, is the question that decides which of the two fixes a real instrument should use.