Circuit Response
every model has an edge, and the figure carries it
Series

Bridge — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A 350 Ω bridge, with ideal leads. The straight line is the expression every textbook gives, Vδ/4; the curve is the solve. They part company at half the fractional change — 0.498% at 1.0% — so the tangent is worth one per cent only up to 2.020%. Driving the bridge from a current source instead halves the departure at every point and moves that edge to 4.040%.

    The bridge that is linear near one point

    The expression for a Wheatstone bridge's output is V delta over four, and the solve says it is V delta over two times two plus delta — low by half the fractional change, exactly, at every change tested. So the tangent is worth one per cent only up to a two per cent change and a tenth of a per cent only up to two tenths. Driving the bridge from a current source instead halves the departure at every point and doubles both edges, which is a change of one component and no change at all to the four resistors being measured.

    part 1 · applied
  2. A tenth of a per cent of lead is 999 µε of strain that is not there. computed by solving, not by drawing. A 350 Ω quarter bridge at zero load, against the resistance in each lead. With both leads in the changing arm the output is 4.99500 mV at 350 mΩ — an apparent fractional change of 0.1998%, which at a gauge factor of 2 is 999 microstrain. With one lead in that arm and one in the arm beside it the output is zero to the last bit, at every lead resistance drawn. The lower panel is what the second arrangement costs: the sensitivity falls as 1/(1 + Rlead/R), which is 0.100% at the same lead and is a calibration constant rather than a drift.

    The leads that are in the bridge

    A strain gauge on the end of two long wires cannot be told from a strain gauge under load: both leads in the changing arm is bit for bit the same netlist as a quarter bridge whose fractional change is larger by twice the lead over the gauge, which at 350 milliohms on a 350 ohm gauge is 999 microstrain that is not there. Moving one of those leads into the arm beside it leaves the output at exactly zero for every lead resistance drawn, and turns a twenty-kelvin drift of 76.8 microstrain into 0.077. That factor is two over the strain being read — 999 at a thousand microstrain and 9990 at two hundred — and it costs 0.100 per cent of sensitivity.

    part 2 · applied
  3. Two leads nobody counts cost 1995 parts per million, and two more leads remove all of it. computed by solving, not by drawing. The error in the reported strain against the resistance in each of the two excitation leads, at 1000 µε on a quarter bridge of 350 Ω excited at 10 V. With four wires the instrument takes the excitation to be the supply's voltage, so every reading is scaled by R/(R + 2r): 1995 parts per million at 0.35 Ω, 54029 at 10. Two more leads brought back from the bridge's own terminals, carrying only the instrument's input current, leave 3.5e-4 parts per million. Exciting with a current instead of a voltage does the same thing with no extra leads at all, because the lead resistance is in series with a source that does not care.

    The two leads nobody counts

    The leads that are in the bridge moved a lead out of the changing arm and turned a 999-microstrain error into nothing. The two leads carrying the excitation are still there, and they scale every reading: 0.35 ohms each on a 350-ohm bridge is 1,995 parts per million, exactly −2r/(R + 2r), the same on a full bridge as on a quarter one, and drifting 0.153 microstrain over twenty kelvin — twice what the three-wire fix left behind. Two more wires brought back from the bridge's own terminals leave 0.00035 parts per million. So does exciting the bridge with a current, which needs no extra wires at all.

    part 3 · applied
  4. A cable's carrier limit is its conductors' matching, not its capacitance. computed by solving, not by drawing. A 350 Ω quarter bridge at 1000 µε, excited at a carrier frequency, with 10 metres of cable at 100 pF a metre on each output terminal. Each terminal sees the two arms meeting there in parallel against that capacitance, so each lags by arctan(2πf·(r/2)·C) with a corner at 909 kHz — and with the two conductors matched the lag CANCELS in the difference: -0.117° where each terminal has -5.71°. A mismatch does not cancel, and it arrives multiplied by the ratio of the excitation to the signal, which at 1000 µε is 2000: at 1 per cent the phase is 40.3° at a hundred kilohertz. So the excitation a 10-metre cable allows for 200 parts per million falls from 205 kHz matched to 1.82 kHz at 1 per cent. And it depends on the strain being measured — 183 Hz at 100 µε, 547 Hz at 300 µε, 1.82 kHz at 1000 µε, 5.45 kHz at 3000 µε — because the amplification is the reciprocal of the signal.

    The carrier a cable allows

    A carrier-frequency amplifier removes an instrument's offset drift by exciting the bridge with alternating current, and the cable that was a resistance at direct current becomes a capacitance. Both output terminals see the same arms and the same capacitance, so the lag cancels in the difference: ten metres matched leaves 0.15 degrees at a hundred kilohertz where each terminal has 6.3. What does not cancel is the mismatch between the two conductors, arriving multiplied by the ratio of the excitation to the signal — two thousand at a thousand microstrain — so one per cent of it takes the allowed carrier from 205 kilohertz to 1.8.

    part 4 · applied
  5. A tenth of a per cent of screen mismatch on the gauge's side cancels the pair's own phase: 18.2 kHz becomes 143 kHz. computed by solving, not by drawing. A 350 Ω quarter bridge at 1000 µε read through ten metres of screened pair: 100 pF a metre from each conductor to the screen, Cs, and 50 pF a metre between the two, Cp. The loss of the synchronously demodulated reading, 1 − cos φ, against the excitation frequency: for the capacitance between the conductors alone, for the screen capacitances alone with a mismatch of 0.1% on the gauge's side, for both together, and for a matched screen with nothing between. For a 200 ppm budget they allow 18.2 kHz, 18.3 kHz, 143 kHz and 205 kHz. The pair's capacitance lags the reading; extra screen capacitance on the gauge's own terminal leads it, by an amount divided by the signal, so at a mismatch of δ·Cp/Cs the two cancel to first order.

    The capacitance that is already a difference

    A screened pair has capacitance from each conductor to the screen, which cancels in a bridge's difference when the two match, and capacitance between the conductors, which is a difference from the start and cannot cancel. Ten metres of it between a 350 Ω quarter bridge and its carrier amplifier allows 18.2 kHz for a 200 ppm loss, against 205 kHz for the screen alone. The two phases are ωR·ΔC/δ and ωR·C, the same slope with one divided by the signal, so a screen mismatch of δ·Cp/Cs on the gauge's own terminal cancels the pair's lag exactly: 0.10% at 1000 µε takes the limit to 143 kHz. That is what a carrier amplifier's capacitance balance finds when it nulls the quadrature — and because the mismatch's phase changes sign with the strain and the pair's does not, the balance holds in tension above half the strain it was set at and never in compression.

    part 5 · applied
  6. Calibrated at the carrier, a matched cable allows 121 kHz where its phase allowed 16.5 kHz. computed by solving, not by drawing. A 350 Ω quarter bridge read through ten metres of screened pair, excited over leads of 0.35 Ω and 3 µH each with 500 pF between them. Solid: the worst error of a synchronous reading over strains from −2000 to +2000 µε after a calibration at +1000 µε at the operating carrier, with the reference taken from the source at the instrument; dashed, from the bridge's own terminals on sense wires; dotted, the phase measure 1 − cos φ the earlier readings of this cable used. For a 200 ppm budget: 16.5 kHz by phase, 119 kHz calibrated against the source and 121 kHz calibrated against the bridge.

    What a calibration at the carrier removes

    The phase a cable puts on a carrier-excited bridge is a loss of 1 − cos φ, and a loss that is the same at every strain is a gain error — which a shunt calibration at the operating carrier exists to remove. Calibrated there, a synchronous reading of a quarter bridge through ten metres of matched screened pair allows 121 kHz for 200 ppm over ±2000 µε, where the phase measure allowed 16.5; the capacitance between the conductors, which no matching could improve, stops mattering. Where the reference is taken — at the instrument or from sense wires at the bridge — moves that by 1.2 per cent. What calibration cannot touch is a screen mismatch: its leak is a fixed voltage, not a fixed gain, and a hundredth of a per cent of it takes 121 kHz back to 11.5.

    part 6 · applied
  7. A 1% mismatch in the cable reads as 3.0% of full-scale displacement, at every frequency. computed by solving, not by drawing. A capacitive displacement sensor — two 10 pF plates driven at ±5 V in antiphase facing a common electrode — read by a charge amplifier through two metres of three-conductor cable with 30 pF a metre between each excitation conductor and the sense conductor. With the two cross-capacitances mismatched by 1 per cent and the sensor centred, the reading's in-phase part (solid) is 3.00 per cent of full scale at every carrier from 100 Hz to 1 MHz: the cable is a sensor capacitance, and its mismatch a displacement. The quadrature part (dashed), which the cable's resistance makes, rises with frequency and a synchronous detector rejects it. With each conductor in its own screen the cross-capacitance is gone and so is the error.

    The sensor that is the capacitance

    Read a capacitive displacement sensor through the same kind of cable that limited a strain-gauge bridge's carrier, and the roles exchange. On the bridge, a 1% mismatch in the cable's capacitance was a quadrature leak that grew as the square of the carrier and could be escaped by a lower one. On a sensor of two 10 pF plates it is 3.0% of full-scale displacement, in phase, at every carrier from 100 Hz to 1 MHz — because a capacitance between an excitation conductor and the sense conductor is in parallel with the measurand and is a sensor capacitance. No detector phase separates it and no carrier escapes it; only geometry does. Screen each conductor separately and there is nothing left to mismatch.

    part 7 · applied
  8. A reference 0° behind the excitation at 20 kHz, after a calibration at +1000 µε: matched, the worst error is 5.79 ppm; with a tenth of a per cent of screen mismatch, 0.607%. computed by solving, not by drawing: the error of a synchronous reading of a 350 Ω quarter bridge through ten metres of screened pair, against the strain read, with the reference 0° from the excitation at the bridge and a shunt calibration at +1000 µε through the same reference, for a matched screen (solid) and for 0.01 and 0.1 per cent of mismatch. The calibration takes out cos(φ − θ), because it multiplies the signal at every strain. It cannot take out a voltage that does not scale with the strain, and the mismatch's leak is such a voltage: a hyperbola through zero at the calibration strain. Worst over ±2000 µε: 0%: 5.79 ppm, 0.01%: 606 ppm, 0.1%: 0.607%.

    The reference that should be late

    A carrier amplifier's reference is taken from the excitation because that is what the output is meant to be in phase with, and nothing says that is where it belongs. A screen mismatch's leak arrives late by the common-mode node's lag and the difference's together, 525 ns on ten metres of pair and a 350 Ω bridge, so a reference on the excitation reads sin 3.7° of it at 20 kHz: 0.607 per cent of error at a tenth of a per cent of mismatch, and 72.6 parts per million with the reference 3.7° late. The null is a delay rather than a phase and does not move with the mismatch, and a delayed reference allows about five times the carrier on any mismatched screen. How exactly it must be held narrows as mismatch times carrier.

    part 8 · applied

All series