The reference that should be late

What a calibration at the carrier removes found that a shunt calibration through the cable takes away the whole of a carrier amplifier’s phase loss, because that loss is the same at every strain, and leaves only the things that do not scale with the strain. It ended on a warning it did not price. A detector reads $I\cos\theta + Q\sin\theta$, so its phase error turns the output’s quadrature into the reading, and the quadrature here is large. A reference a tenth of a degree out, it guessed, would multiply a large thing by a small one and become the next limit.

That guess carries an assumption no earlier essay in this sequence examined: the correct reference is the excitation, and any departure from it is an error. The output of a bridge is meant to be in phase with its excitation, so that is where every amplifier takes its reference, and the question has always been how closely it holds. Sweep the reference’s phase instead of assuming it, and the best place for it turns out to be somewhere else. The error falls by a factor of eighty when the reference is moved a little under four degrees later, and the phase it wants is a property of the cable.

A reading with a rotation in it #

The circuit is the earlier essay’s: a 350 Ω quarter bridge on 10 V of excitation, read through ten metres of pair with 100 pF per metre from each conductor to the screen and 50 pF per metre between the two. A reading is the in-phase part of the output against a reference, and here the reference is $\theta$ behind the excitation as the bridge’s own terminals see it. The sense wires take care of the excitation leads, whose lag the earlier measurement showed to be worth 1.2 per cent of the allowed carrier, and which this essay leaves out of the argument.

The calibration passes through the same reference. A shunt across one arm simulates $+1000$ µε, and the amplifier scales its reading to that, at this carrier and at this reference phase. So a reference error does not appear as a gain error at all: any fixed rotation multiplies the signal by $\cos(\varphi - \theta)$ at every strain, the calibration measures the product, and the product is divided out. That is the part of the warning that does not survive. A phase error in the reference is, on its own, exactly the kind of thing a calibration exists to remove.

What it cannot remove is anything in the output that does not scale. There are two such things, and they are different sizes. They want different references, and the excitation is not the right one for either.

The leak has a phase of its own #

The large one is the leak. The carrier a cable allows found that a screen mismatch turns the bridge’s five volts of common mode into a difference, and what a calibration at the carrier removes identified it as a fixed voltage, the same number of microvolts at every strain, and therefore a hyperbola through zero at the calibration strain. A voltage formed by current through a capacitance is close to quadrature with the voltage driving it, which is why the phrase quadrature leak has been used throughout, and why a detector on the excitation is expected to reject it.

Close to quadrature is not quadrature. It is close by the angle a real time constant makes at this carrier, and at 20 kHz that angle is nearly four degrees. So the reference on the excitation reads $\sin 3.7°$ of the leak, about 6.5 per cent of a voltage that at a tenth of a per cent of mismatch is about a tenth of the signal at the worst strain. The result is the 0.607 per cent in the opening figure.

The figure sweeps the reference through twelve degrees for five mismatches, and each mismatched curve is a deep V with its point at the same place: 3.70°, for 0.01 per cent and for 0.3 per cent alike. That is a stronger fact than a good setting would have been. If the null moved with the mismatch, finding it would mean measuring a mismatch. It does not move, so the reference’s best setting is a property of the circuit, and it applies to a cable whose mismatch nobody knows.

At the null the error is 72.6 parts per million at 0.1 per cent of mismatch, against 0.607 per cent at the excitation, a factor of 84. Going the other way hurts: a reference a degree ahead of the excitation gives 0.773 per cent. A designer who reads the fixed phase error of a detector as a symmetric tolerance around the excitation has, at this carrier, an asymmetric result to discover. A small lag improves the reading and a small lead worsens it, and the sign of the phase error decides which.

A delay and not a phase #

A setting that holds at one carrier would be a curiosity. Repeated at six carriers from 2 to 60 kHz with the mismatch at a tenth of a per cent, the null’s phase rises in strict proportion to the carrier, which is to say it is a delay.

The size of the delay is a sum of two time constants that can be read off the schematic. The five volts of common mode reach each output terminal through the bridge’s arms, 175 Ω in parallel from each terminal, against that terminal’s own capacitance to the screen: 1 nF, so $\tau_c = 175$ ns. Where the capacitances differ, the difference between the two terminals is then a voltage across the output, and it sees the differential time constant, 350 Ω across the pair’s capacitance and half the screen’s: $\tau_d = 350$ ns. A leak passes through both, one after the other. Together they are 525 ns.

The solve puts the null about 3 per cent earlier than that: 3.66° against 3.78° at 20 kHz, and in the same proportion at the other carriers. Part of the shortfall is that the phase of a ratio of two small numbers is not quite the sum of their first-order lags, and part is that the worst error across several strains has its least value slightly away from the leak’s exact quadrature, since a matched screen’s own residual is folded in. A law that lands 3 per cent high and in proportion at every carrier is doing its job: it says what to build, and that the true value is a trim of a few per cent away from it.

The physical reading is more useful than the formula. A voltage on the excitation takes 525 ns to come out of the screen’s capacitance as a difference. A detector that wants to reject it has to look at what the excitation was 525 ns ago, because the detector’s job is to be exactly at quadrature to what is arriving and not to what was sent.

The matched screen’s null is somewhere else #

The small thing that does not scale deserves its own figure, because it has the same form and a different answer. With no mismatch at all, the earlier essay found the calibrated limit at 121 kHz, because the lag itself changes a little with strain: the gauge’s resistance moves, so the resistance the pair’s capacitance works against moves too.

The natural guess is that the best reference for this is the output itself, since the output’s phase is what is changing and a detector on it should be first-order blind to that. The guess is wrong by a factor of two, and the reason is worth a line. A single pole’s output is not $e^{-j\varphi}$ but $\cos\varphi\,e^{-j\varphi}$, because its magnitude falls with the same lag that moves its phase. So the reading is $S\cos\varphi\cos(\varphi - \theta)$, and its sensitivity to a change in $\varphi$ is $-\sin(2\varphi - \theta)$, which vanishes at $\theta = 2\varphi$. A reference on the output is halfway there. Removing the capacitance to the screen, so that one pole is all there is, the solve puts the null at 12.5° against twice the lag’s 12.6, and at 18.7 and 24.8 against 18.7 and 24.8.

With the screen’s capacitance present the matched null moves further still, to about 2.6 times the lag, because the strain then also unbalances the common-mode route. Nothing about that needs fixing. The number to take from it is that at 150 kHz a matched bridge’s error is 300 ppm on the excitation and 2.06 at the right reference, a factor of nearly 150. And the phase is not the leak’s: at 150 kHz, the leak’s delay asks for 28.4°, and the matched null is at 24.6°. They are close enough that one delay serves both tolerably, which is what the last figure below tests.

How narrow the window is #

A null is a promise of exactness, and the first question to ask of one is how exactly it has to be held. The error beside the null is the leak’s share times the reference’s distance from it, so the window for a stated budget is that budget over the leak’s share, and the share grows with both the mismatch and the carrier.

The lines are as straight as the argument predicts. At 20 kHz the reference may stray ±1.20° at 0.01 per cent of mismatch, ±0.120° at 0.1 per cent and ±0.0120° at 1 per cent. At 5 kHz each window is exactly four times wider. So the product of mismatch, carrier and phase tolerance is a constant for this budget, which is the original warning given its size at last: at 0.1 per cent of mismatch and 20 kHz, a detector needs an eighth of a degree around the right phase. A tenth of a degree is well within what careful analogue multipliers and any digital reference can do, so the warning holds and it is survivable.

It is also a warning about 1 per cent of mismatch, where ±0.012° at 20 kHz is a precision no analogue detector holds, across temperature and between units. That result joins the other reasons to avoid such a cable at that carrier. The capacitance that is already a difference reached a similar place by a different route: there are parts of this problem that a measurement cannot be set up to escape. In ±0.0120°, the part that no setting escapes is the mismatch.

Three references, one bridge #

The comparison that decides whether any of this is worth building takes each candidate reference and asks for its allowed carrier, across the mismatches a real cable might have.

The excitation is the baseline: 121 kHz matched and 3.62 kHz at a tenth of a per cent. The second candidate is what a phase trim on a carrier amplifier does, adjusting the reference until the output’s quadrature reads zero at the calibration strain. It gains about a quarter on a matched screen, 156 kHz, and in the middle of the range, and at 0.1 per cent it has given it all back: 3.62 kHz, exactly level. Beyond that it loses, reaching 565 Hz at 1 per cent against the excitation’s 1.14 kHz. The trim is a reasonable-looking procedure with a defect a sweep exposes at once. It nulls the output’s quadrature, and with a mismatched screen that quadrature is largely the leak’s, so the reference is being steered by the size of the very defect it should be ignoring.

The third candidate is simply late by 525 ns, at every carrier, with nothing adjusted against the output. It allows 256 kHz matched, 59.3 kHz at 0.01 per cent, 19.7 kHz at 0.1 per cent and 6.58 kHz at 1 per cent: about five times the excitation’s carrier on any mismatched screen, and twice it on a matched one. With the delay two per cent out it still allows 15.6 kHz at 0.1 per cent. It is a fixed setting, and the only thing it needs to know is the cable: length, capacitance per metre and bridge resistance. A carrier amplifier has those values at setup.

What a delay of half a microsecond means in practice #

A delay is cheap to build at this scale, whether as a small RC on a reference comparator or as one sample period in a digital demodulator running at a couple of megahertz. Its trouble is not its construction but its validity, because the 525 ns is a sum over the cable and the arms. A ten-metre cable has it; a hundred-metre cable would have ten times as much and has already been ruled out at this carrier by other essays. A half bridge with equal arms has the same $\tau_c$ and $\tau_d$, and a full bridge has too, so the setting is transferable across bridge wiring at the same arm resistance. A 120 Ω gauge divides it by nearly three.

So a measurement system that reads its cable’s capacitance at setup, with a capacitance meter or from a step, could set the delay within the 2 per cent the figure tolerates, since a 2 per cent capacitance measurement is not hard. Nothing in the argument requires knowing the mismatch, which is the thing nobody can measure from one end of a cable without a bridge at the far end. Its absence is why the window figure, which depends on the mismatch, is a limit on the size of leak that can be survived rather than a setting.

There is also a cost of a sort. The late reference is not in phase with the signal at the bridge, so it reads the signal through $\cos(\varphi - \theta)$: at 20 kHz, with the output lagging by a little over a degree and the reference by 3.8°, it loses about one part in a thousand of signal, which the calibration removes and the noise floor barely notices. There is no comparable effect on a bridge’s linearity: the bridge that is linear near one point is about a gauge and not about a phase, and the two leads nobody counts concern the excitation’s size and not its timing.

Why nobody draws this reference #

Part of the reason is that the leak has usually been treated as a pure quadrature term, since at low carriers it nearly is, and a detector’s job has been stated as rejecting quadrature. At 1 kHz the delay is a fifth of a degree and the excitation’s reference reads only $\sin 0.19°$ of it, so a designer working at low carriers has no reason to notice. At 20 kHz, the frequency carrier amplifiers moved to in order to avoid low-frequency drift, the same physics has become a fivefold penalty in carrier.

The other part is that the common-mode rejection specification hides it. The rejection four resistors decide measures rejection as a ratio of magnitudes, and a leak expressed that way has no phase at all. Asked to null a leak, a designer balances the capacitances, which is expensive; asked for the phase of the leak, the same designer would notice that a much cheaper thing is available. The sensor that is the capacitance found the situation reversed for a capacitive sensor, where the leak becomes in phase with the signal and no reference can reject it. That case makes the point from the other side: for the resistive bridge, the phase of the leak is the whole reason a detector works at all, and it deserves to be measured.

What the model leaves out #

The cable here is lumped, ten metres long and well under a wavelength at any carrier on these figures, and the arms are pure resistances. An ordinary foil gauge has a little inductance and a little capacitance to its substrate, both of which add to the leak’s phase by amounts that depend on the installation. The mismatch here is a difference in the capacitance to the screen, and is the same at all frequencies. A dielectric whose loss differs between conductors would add a small in-phase component to the leak, and an in-phase leak is the one kind of leak no reference can remove. That is the term that sets how deep the null really is, and on these figures it is absent, so the factor of 84 is an upper bound on what the null gives.

The detector is an ideal multiplier with an ideal low-pass. A real square-wave reference also reads the output at the odd harmonics of the carrier, which have their own phases and their own leak, and whether a delay that is right for the fundamental is right for the third is not computed here.

The numbers worth keeping #

A 350 Ω bridge on ten metres of pair reads a screen mismatch’s leak through $\sin(\omega(\tau_c + \tau_d))$, with $\tau_c + \tau_d = 525$ ns from the schematic. At 20 kHz and 0.1 per cent of mismatch the error is 0.607 per cent on the excitation and 72.6 parts per million at 3.70°, and the null stays there for any mismatch and scales with the carrier. A reference delayed by that much allows 19.7 kHz at 0.1 per cent against the excitation’s 3.62 and a phase-trimmed reference’s 3.62, and it keeps most of that with its delay 2 per cent out. The window around the null at that mismatch and carrier is ±0.120°. The excitation is where a detector’s reference has always been taken, and on a resistive bridge at a high carrier it is about half a microsecond too early.

Still open: the third harmonic of a square reference, the half bridge, and a delay that measures itself #

A square-wave reference. A detector that switches rather than multiplies reads the output’s odd harmonics at weights of a third, a fifth and so on, and the leak grows with frequency, so the third’s contribution is not negligible even though its weight is small. Each harmonic has its own null at the same delay, since a delay is a delay at every frequency, but the second-order departures are not the same, and whether the deep null survives a switching detector is the measurement the ideal multiplier here does not make.

A half bridge. With two active arms in opposite arms of the output, the signal doubles and the leak does not change, so every window in this essay should open by a factor of two. The pair’s residual depends on how each terminal’s resistance moves, and with equal and opposite changes it may cancel altogether. That would leave the leak as the only thing the reference has to deal with.

A delay read from the reference’s own quadrature. Sweeping the reference across a null and finding the minimum of a quadrature reading at zero strain would locate the delay on a bench, without knowing the cable. A bridge at balance outputs nothing but its leak, so its quadrature zero is exactly the setting this essay computes. Whether that procedure survives a screen mismatch small enough to give little to read, and what noise sets the accuracy of the sweep, is the commissioning question this essay leaves.