The carrier a cable allows

The two leads nobody counts ends by naming three things it does not solve, and the second is a question rather than a caveat:

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.

The answer is not a property of the cable’s capacitance, which is the first thing the measurement says.

What a phase shift costs, and why it is second order #

A carrier-frequency amplifier excites the bridge with a sinusoid, amplifies the bridge’s output with an amplifier whose own offset is irrelevant because it is not at the carrier frequency, and then recovers the strain by multiplying the amplified output by the excitation and taking the mean. That last step is a phase-sensitive detector, and what it returns is the component of the output in phase with the reference.

So a phase shift $\varphi$ between the output and the reference costs $\cos\varphi$ of the reading. That is second order in the phase — a degree costs 152 parts per million, and a tenth of a degree costs 1.5 — which is why the arrangement tolerates a great deal of phase and why the question above has an answer at all.

The budget it is spent against is the thing the arrangement was chosen for. A direct-current instrument’s offset drift is a few hundred parts per million of a bridge’s output over a working temperature range — the leads that are in the bridge prices one contribution to it at 76.8 microstrain of drift over twenty kelvin before the three-wire fix and 0.077 after — and if the carrier’s own phase error costs more than that, the arrangement has bought nothing.

Two hundred parts per million is therefore the natural budget, and it corresponds to a phase of 1.15 degrees.

What the alternating excitation was bought for #

It is worth being explicit about what is on the other side of this ledger, because the whole question is whether the phase error costs more than the drift, and the drift is the thing three essays of this sequence of essays have been measuring.

Amplifier offset drift. A direct-current instrument amplifies a five-millivolt signal by two hundred, so its own input offset appears in the reading directly. A good part drifts a few tenths of a microvolt per kelvin, which over a twenty-kelvin working range is a few microvolts on five millivolts — hundreds of parts per million. A carrier-frequency arrangement puts the signal at a frequency where the amplifier’s offset is not, so the offset is rejected outright rather than trimmed.

Thermoelectric voltages. Every junction of two dissimilar metals in the signal path is a thermocouple of tens of microvolts per kelvin, and a bridge’s wiring has several. Those are direct-current sources too and the carrier rejects them for the same reason.

And low-frequency noise. The amplifier’s own flicker noise rises towards direct current without a floor, which the corner where averaging stops working measures: a longer average narrows the bandwidth and the density under it rises to meet it. Moving the signal to a carrier moves it above the flicker corner, where averaging works again.

Against those three, the arrangement’s costs are the phase error measured here and the complication of generating and detecting a carrier. The first two of the three benefits are hundreds of parts per million and the third is a shape rather than a number — so the two hundred parts per million this essay budgets against is the right order and the comparison is genuinely close.

Which is the useful conclusion and it is a narrow one. A carrier-frequency bridge with a well-matched cable is clearly better than a direct-current one, and with a badly matched cable it is clearly worse, and the deciding parameter is not in either instrument’s specification. It is in the cable’s, and cables are specified by capacitance per metre rather than by the matching between conductors.

What cancels #

Each output terminal sees the two bridge arms that meet there in parallel — 175 ohms, half the arm resistance — loaded by whatever capacitance the cable presents. That is a first-order lag with a corner at

$$f = \frac{1}{2\pi (R/2) C}$$

which for ten metres of a hundred picofarads a metre is 909 kilohertz — the same ratio of a resistance to a reactance that the millivolts in the wire finds a shared return conductor’s corner at, arriving here with the two quantities exchanged. At a hundred kilohertz each terminal lags by 6.3 degrees, which by the arithmetic above would cost six thousand parts per million — thirty times the budget.

It costs nothing, because both terminals lag identically. The measurement is the difference between them, and a common lag subtracts out exactly as a common voltage does: the difference lags by 0.153 degrees at a hundred kilohertz, which is a fortieth of each terminal’s own and costs 3.6 parts per million.

That is the same cancellation one voltage added to two readings measures for a shared return conductor, and it has the same character: exact by topology rather than good by tolerance, and limited by whatever the two paths do not have in common.

So a ten-metre cable with perfectly matched conductors allows 205 kilohertz of carrier against a two-hundred-part-per-million budget, which is more than any carrier-frequency amplifier uses.

What does not, and the factor it arrives with #

The two conductors are not matched. They are two wires in one cable, of slightly different length, with slightly different insulation and slightly different distances to the screen, and a per cent between them is unremarkable.

A mismatch converts the common lag into a differential one, and it arrives multiplied. The excitation is ten volts and the signal is five millivolts, so the common-mode quantity being converted is two thousand times the differential quantity it is being converted into — and a one per cent conversion of something two thousand times larger is twenty times the signal’s own phase behaviour.

Exactly inversely proportional to the mismatch, over two decades of it. A ten-metre cable whose two conductors match to one per cent allows 1.8 kilohertz of carrier, which is below the three or five kilohertz such instruments commonly use — so the phase error is already costing more than the offset drift the arrangement was built to remove.

This is the mechanism the corner the instrument has no part in measures on the other side of the same bridge: a 95-decibel instrument falling twenty decibels a decade above 290 hertz when its source has a kilohm of imbalance and ten picofarads at each input, with the mechanism being a difference of two time constants rather than anything about the part. Here it is a difference of two time constants again, in the cable rather than at the amplifier’s inputs, and amplified by the smallness of the thing being measured rather than by a resistor tolerance.

The strain is in the answer #

The amplification is the ratio of the excitation to the signal, and the signal is the strain. So the carrier a cable allows depends on how much strain is being measured, which is not a dependence anybody would look for in a cable specification.

Ten times the strain, ten times the carrier. Which inverts the usual reading of a measurement’s difficulty: the small readings, which are the ones a designer worries about because they are near the noise floor, are also the ones where the cable’s phase error is largest — and it is largest in proportion, so it does not average away and does not fall with the signal.

The practical form of that is blunt and worth stating. A carrier-frequency bridge calibrated at full scale is not calibrated at a tenth of it. The error is a gain error rather than an offset — the reading is multiplied by $\cos\varphi$ — and $\varphi$ itself grows as the signal shrinks, so the instrument’s gain is a function of what it is reading. That is a nonlinearity, and it is not one that any of the three essays below would find: the bridge that is linear near one point measures a departure of half the fractional change, which is a property of the bridge’s own arithmetic and is present at direct current — and which the bridge that is linear near one point halves by exciting with a current instead, a repair that does nothing whatever for this one.

The measurement that does not see it #

One more property of this error makes it hard to find, and it is the reason a bench does not catch it.

The bridge’s output magnitude is barely affected. At the frequency where the in-phase loss reaches two hundred parts per million, the magnitude is out by a few parts per million — because a phase shift moves a phasor round a circle and hardly changes its length. So an instrument that rectifies its output rather than multiplying it by a reference reads almost the right answer, and one that multiplies reads the wrong one.

Three consequences.

The better detector has the worse error. A phase-sensitive detector is chosen over a rectifier because it rejects everything not at the carrier frequency, which is most of the noise and all of the interference. It is that same selectivity — reading one component of a vector rather than its length — which makes it sensitive to a phase the rectifier ignores.

A quadrature reading is available and is usually discarded. The detector that produces the in-phase component can produce the quadrature one from the same multiplication with the reference shifted ninety degrees, and the ratio of the two is $\tan\varphi$. So the error is directly measurable by the instrument that suffers from it, at no extra hardware, and the correction is a division by $\cos(\arctan(Q/I))$ — which is to say, the magnitude, which is what the rectifier was reading all along.

And the quadrature channel is where the cable’s mismatch shows up as a number. Watching it while the carrier frequency is swept gives the mismatch directly, since the quadrature component is proportional to the frequency below the corner. That is a calibration a carrier-frequency instrument could perform on itself and, as far as anything here shows, does not.

Where the mismatch comes from, and whether it can be trimmed #

The number the arrangement turns on is a per cent between two pieces of wire, so it is worth asking what sets it and what could be done about it.

Length. Two conductors in a cable are not the same length: one lies on the outside of every bend. In a tightly twisted pair the difference is a fraction of a per cent over ten metres; in a loose multicore it can be several.

Position relative to the screen. The capacitance to the screen depends on the distance to it, and in a multicore cable two conductors sit at different radii. This is the largest term in a cable not built as a pair, and it is why a twisted pair inside a screen is the standard arrangement rather than two wires in a bundle.

And whatever else is on the terminals. The cable’s capacitance is in parallel with the amplifier’s own input capacitance, the connector’s, and the board’s — so the mismatch is between two totals, and a cable matched to a tenth of a per cent connected through a connector whose two pins differ by a picofarad has the connector’s mismatch rather than the cable’s.

That last one decides whether a trim is possible, and it is: a small adjustable capacitance across the better-matched terminal brings the two totals together. The quantity to trim against is the quadrature component of the detector’s own output, which is proportional to the residual mismatch and is available without any extra measurement — so the adjustment has a null to trim to rather than a number to aim at.

Which is the same structure as that essay’s own repair. The two leads nobody counts removes an excitation-lead error by bringing two more wires back from the bridge’s own terminals, converting a quantity that must be small into one that must be equal — and here the conversion is from a capacitance that must be small to two capacitances that must match, with a trim available because the instrument can see its own residual.

The reason to prefer the trim to a better cable is arithmetic rather than economy: the mismatch is amplified by the reciprocal of the strain, so a fixed cable specification is a carrier limit that falls as the readings get smaller, and a trim against a null is not.

What this does not settle #

One capacitance per terminal. A real cable has capacitance between its conductors as well as from each to the screen, and the conductor-to-conductor term acts differentially rather than in common — so it does not cancel even when the two conductors match. Its size is a fraction of the to-screen capacitance and its effect is not measured here.

The excitation leads are still resistive. That essay’s whole subject is the two leads carrying the excitation, scaling every reading by $-2r/(R+2r)$. Those leads have capacitance too, and at a carrier frequency it loads the excitation source rather than the output — which is an amplitude error on the excitation, and if the detector’s reference is taken from the excitation at the instrument rather than at the bridge, it does not divide out.

And nothing here is about the gauge. A strain gauge is a resistance with a small inductance and a small capacitance of its own, and at a carrier frequency of a few kilohertz neither matters. At the two hundred kilohertz a matched cable would allow, they might.

Still open: the differential capacitance, and the reference taken at the wrong end #

The conductor-to-conductor term. It is differential by construction, so it survives the cancellation that removes the common one and its phase is not amplified by the signal’s smallness — which makes it a smaller effect with a different scaling, and the crossover between the two is the quantity a cable specification would need. A shielded twisted pair quotes both capacitances and this measurement uses one of them.

Where the reference is taken. Everything above assumes the detector’s reference is the excitation as the bridge sees it. If it is taken at the instrument instead, the excitation leads’ own phase shift appears in the reference rather than in the signal and adds directly — and six-wire sensing, which the essay before it shows removes the excitation leads’ resistance, would remove this too if the sense wires carried the reference. Whether they do is an arrangement question with a measurable answer.

And the same arithmetic for a capacitive sensor. A displacement sensor is a capacitance read by exactly this arrangement, with the roles of the bridge’s resistances and the cable’s capacitance exchanged. The amplification by the signal’s smallness should be the same; what changes is which of the two is the sensor and which is the parasitic, and that inversion is worth drawing because the conclusion about cable matching ought to survive it.

What is checked #

The cancellation is stated as a ratio, not as a level: the difference’s phase is required to be under a twentieth of each terminal’s own, which is the claim that the lag is common and that the bridge reads what is between the terminals.

The amplification is required in the other direction — with a mismatch, the difference’s phase must exceed the matched case’s by more than twenty times — because the essay’s finding is a pair and either half alone reads as a reassurance or as an alarm.

The allowed carrier is bisected on the solved network at every setting, and required to be inversely proportional to the mismatch across the slider, which is what says the mismatch is the variable rather than the capacitance.

And the phase is measured against the output’s own low-frequency value rather than against zero. A quarter bridge’s differential output is negative — the changing arm pulls its node down — so an absolute angle is a hundred and eighty degrees plus the shift, and the first version of this reported 199 degrees of error against a closed form that was right.