What a calibration at the carrier removes

The capacitance that is already a difference found that a screened pair’s capacitance between its two conductors limits a carrier-excited strain-gauge bridge far more than its capacitance to the screen: ten metres allowed 18.2 kHz for two hundred parts per million, and no matching of the conductors could raise it, because that capacitance sits across the output and is already a difference. It closed by naming a question it could not answer from where it stood. Every phase it measured was against the excitation as the bridge sees it; a detector that takes its reference at the instrument sees the excitation leads’ own lag as well, and whether that lag adds to the pair’s phase or can be used to cancel it is decided by where the reference is taken.

Answering that question needs the excitation leads in the circuit, and it needs the detector to be a detector rather than a phase. Both turn out to matter less than the thing the question took for granted: that $1 - \cos\varphi$ is what the reading loses.

The loss that is a gain #

Every essay in this sequence has scored a cable by $1 - \cos\varphi$, the fraction by which a synchronous detector’s in-phase reading falls short when the output lags its reference by $\varphi$. That is a correct account of what the detector reads. What it leaves unsaid is that it is a gain error: it multiplies the reading, and if the lag is the same at every strain, the reading at every strain is short by the same fraction.

A gain error that does not depend on the quantity being measured is exactly what calibration removes. A strain-gauge amplifier is calibrated by a shunt — a known resistor switched across one arm, simulating a known strain — and a carrier amplifier does it at the carrier, through the same cable, so the calibration sees the same lag as the measurement and scales it away. After a shunt calibration the question is no longer how large the lag is. It is how much of the reading does not scale with the strain, because that is the only part a gain cannot correct.

With a matched screen, calibrated, the reading stays inside two hundred parts per million up to 121 kHz, over every strain from −2000 to +2000 µε in both directions. The phase measure over the same strains allows 16.5 kHz. The pair’s capacitance — the whole of the earlier limit — is a lag that is nearly the same at every strain, so the calibration takes nearly all of it.

What is left, at 121 kHz, is second order: the gauge’s own resistance changes with the strain, so the resistance the pair’s capacitance works against changes too, and the lag is not quite the same at every strain. Remove the pair capacitance altogether and the limit rises to 231 kHz; that difference is the part of the pair’s lag that varies with strain, and it is all the calibration leaves of it.

The excitation leads, and where the reference is taken #

The excitation now arrives over ten metres of its own: two conductors of 0.35 Ω — the lead resistance the two leads nobody counts worked with — and 3 µH each, with 500 pF between them at the bridge. The bridge presents 350 Ω to the leads, so the excitation at its terminals lags the source by roughly the leads’ inductance over that resistance.

At 20 kHz the lag is 0.126°, a tenth of the pair’s own lag there, and it rises in proportion to frequency, as the pair’s does. So the question the earlier essay posed has a size: taking the reference at the instrument adds about a tenth to the phase that has to be accounted for. By the phase measure that is the difference between 16.5 kHz with the reference at the instrument and 18.2 kHz for the pair alone.

After calibration it is almost nothing. The leads’ lag is also the same at every strain — the leads do not know what the gauge is doing — so the calibration absorbs it along with the pair’s: 119 kHz with the reference at the instrument, 121 with it at the bridge, 1.2 per cent apart. Six-wire sensing, with the reference brought back on the sense wires, earns that 1.2 per cent.

That is a small answer to a question the earlier essay framed as open, and it is the right one to have. Sense wires remain worth having for the reason the two leads nobody counts gave — the excitation leads’ resistance, which drifts with temperature and scales every reading — but not for their phase. A calibration at the carrier does the phase’s work already.

What the calibration cannot touch #

The screen mismatch is different in kind, and the difference is exactly the one the calibration exposes.

A mismatch between the two conductors’ capacitances to the screen does not multiply the signal. It converts the bridge’s common-mode voltage — half the excitation, present on both output terminals whatever the strain — into a differential one, so it adds a fixed voltage to the output: a leak, the same number of microvolts at every strain. As a share of the reading it is larger the smaller the strain, which is why the carrier a cable allows found it amplified by the reciprocal of the signal.

A calibration at one strain folds the leak’s share at that strain into the gain, and it is then wrong everywhere else, by the leak’s share times $(\delta_0/\delta - 1)$: nothing at the calibration strain, twice the share at the same strain of the opposite sign, and without bound towards zero.

The error curve is exactly the shape of a fixed voltage divided by a varying one: odd about zero strain after removing the calibration point, zero at +1000 µε, and a hyperbola either side. The figure checks the shape rather than trusting it — the error at −1000 µε is required to equal the leak read at +500 µε times the factor the hyperbola gives, to five per cent — and it matches, which is the statement that the mismatch’s effect is a fixed voltage and nothing more.

The leak is common-mode rejection by another name #

The leak deserves a second description, because the one above — a screen mismatch converting the bridge’s common mode into a difference — is a statement made before about other instruments, and the earlier arithmetic carries over.

A bridge’s two output terminals sit at half the excitation, five volts of common mode under five millivolts of signal at a thousand microstrain. Anything asymmetric between the two terminals’ paths to the detector turns a fraction of those five volts into a difference. The rejection four resistors decide measures it for a difference amplifier’s resistors, the rejection two capacitors lose at their corner for a pair of coupling capacitors, and the corner the instrument has no part in for a source’s own imbalance meeting an amplifier’s input capacitance — which is this circuit, with the cable in place of the input capacitance. Each found a conversion proportional to the mismatch and rising with frequency, and each found that it cannot be told from signal once it has happened.

What the bridge adds is the ratio. A thousand microstrain is a thousandth of the common mode it rides on, so a conversion of one part in a million of the common mode is a thousandth of the reading — the reciprocal-of-the-signal amplification the earlier essays met. An instrument amplifier quoted at 120 dB of rejection is quoted at a common mode it can reject; a bridge’s cable has no such rating, and its rejection is its match.

Why the phase measure was the right score until now #

It is worth saying why every essay before this one scored the cable by its phase, because the reason is sound and the change here is in the circumstances, not in the arithmetic.

The phase measure answers the question an uncalibrated instrument has to answer: how far is the reading from the truth? A direct-current calibration — a shunt resistor switched in with the excitation steady — cannot see a lag that exists only at the carrier, and an instrument calibrated that way really does read short by $1 - \cos\varphi$. That is the instrument the bridge that is linear near one point and the leads that are in the bridge were built around, and for it the pair capacitance’s 18.2 kHz is the limit.

A carrier amplifier calibrated at its carrier is a different instrument, and for it the question becomes how far does the reading move with the strain, beyond what the strain explains? Every gain the cable applies is absorbed; only the part of the output that is not proportional to the strain survives. The two scores agree whenever the lag is small, and they part company exactly when the lag is the same at every strain — which, for a matched cable, it very nearly is.

How little mismatch undoes it #

Because the calibration removes everything else, the mismatch is the whole of what is left, and it takes very little.

A hundredth of a per cent is a tenth of a picofarad on a ten-metre cable — less than the capacitance a connector adds — and it takes the calibrated limit from 121 kHz to 11.5. The sevenfold gain the calibration found is only available to a cable matched better than any cable is made, and at the mismatches a real screened pair has, a few tenths of a per cent to a few, the calibration wins by a smaller factor: 1.28 at 0.3 per cent, 3.66 at 3.

The middle of the table is the interesting part. Between a hundredth and a tenth of a per cent the phase measure is slightly better than the calibrated one. It is not that calibration hurts; it is that the two measures charge the leak differently at small strains, where it is largest, and at these mismatches the calibration’s advantage on the pair has already been spent.

The balance, reconsidered #

The earlier essay found a way to cancel the pair’s phase without calibration: mismatch the screen deliberately, by $\delta_0C_p/C_s$ on the gauge’s terminal, so that the leak’s phase opposes the pair’s at the strain $\delta_0$. At 1000 µε that is a tenth of a per cent, and it raised the phase measure’s limit from 18.2 to 143 kHz in tension and cut it in compression.

Under calibration that balance is a mistake. It is a tenth of a per cent of mismatch, and the table’s row for 0.1 per cent is 3.62 kHz. The balance worked by using the leak to cancel a gain error, which is the one thing a calibration does better and in both directions; put the two together and the calibration removes the gain error and the balance’s leak remains. Calibrate, and match; do not balance. The capacitance balance belongs to an instrument that cannot calibrate at its carrier, and it buys that instrument one sign of strain.

What a designer takes #

Calibrate at the carrier. A shunt calibration through the cable at the operating frequency removes the pair capacitance’s lag and the excitation leads’ lag together, and for a well-matched cable it multiplies the usable carrier by seven. It costs a switched resistor, which a strain-gauge amplifier already has.

Buy the match, not the capacitance. After calibration the pair capacitance hardly matters and the screen mismatch is everything; a cable with more capacitance per metre and better matching is the better cable. The mismatch sets a floor on the error at small strains that no calibration moves, and a symmetric range down to 300 µε is where it bites.

Take the reference where it is convenient. At the bridge or at the instrument, the difference is 1.2 per cent in carrier. Sense wires earn their place through the excitation leads’ resistance, not their phase, and an instrument that already has them for that reason should use them for the reference too, since the 1.2 per cent is free.

And keep the carrier low if the range goes near zero. The leak’s share grows without bound as the strain falls, so a measurement that has to read small strains in both directions around zero is limited by the mismatch at any carrier, and the remedy is a lower carrier, where the leak is smaller in proportion to the frequency.

How the readings were computed #

Each point is a solved network: the source at the instrument, two excitation conductors each with 0.35 Ω and 3 µH, 500 pF across the bridge’s excitation terminals, the four 350 Ω arms with the gauge’s change $2 \times$ strain, each output conductor’s 100 pF to the screen with any mismatch on the gauge’s side, and 500 pF between the output conductors.

The synchronous reading is the in-phase part of the output against the reference — the source itself, or the excitation at the bridge’s own terminals as sense wires would deliver it. The calibration scales that reading by its ratio to the direct-current reading at +1000 µε, at the same carrier. The error is the calibrated reading at another strain against that strain’s own direct-current reading.

Checked: at the calibration strain the error is zero at every carrier to a part in $10^{12}$, which is the calibration doing what it says; matched, the calibrated limit is required to exceed the phase measure’s by more than five times; the reference’s position is required to change the limit by less than a tenth; and the mismatch’s error is required to follow the hyperbola of a fixed leak.

A perfect shunt, a lumped cable and a clean reference #

The calibration is perfect. A shunt resistor has its own tolerance and its own lead, and it simulates a strain in one arm only; a calibration good to a part in ten thousand is already a good one, and it bounds everything above.

The cable’s capacitances are lumped. At ten metres and a hundred kilohertz the cable is a thirtieth of a wavelength, and treating it as a lumped network is right; at a hundred metres and a megahertz it would not be.

And the detector’s reference is clean. A reference with its own phase noise or a small phase error rotates part of the quadrature output into the reading, and the quadrature output here is large — the pair’s lag and the leak are both mostly in quadrature — so a detector’s phase accuracy is a specification the calibration does not remove.

Still open: the reference’s own phase error, the half bridge, and the sensor that is the capacitance #

A reference that is slightly wrong. After calibration the output’s quadrature part is what a phase error in the reference turns into a reading error, and the pair’s lag is almost all quadrature. So a reference phase error of a tenth of a degree multiplies the pair’s lag into the reading, and the detector’s phase accuracy becomes the next limit — one that sense wires would help with, since they deliver the bridge’s own phase.

A half bridge. With a second active arm the signal doubles while the leak does not, so the mismatch’s share halves and the calibrated limit should rise by the same factor; the pair’s residual, being about how the arms change the source resistance, may not follow. Which of the two sets the calibrated limit on a half bridge is the next measurement.

The capacitive sensor. When the sensor is itself a capacitance read by this arrangement, the cable’s capacitances are in parallel with the thing being measured rather than across a resistance, and their mismatch stops being a quadrature leak and becomes a displacement in phase with the signal. What calibration can do about that, and what a screen can, is the inversion the whole of this sequence has been approaching.