The bridge that is linear near one point

The Wheatstone bridge is the oldest circuit in this collection and it is still the standard way of turning a small change in a resistance into a voltage. Its design expression is one line long, it is correct to first order, and this essay measures the second order — which turns out to be exactly half the fractional change and therefore to arrive much sooner than anybody expects.

What the solve says #

Four resistors, one of them $R(1+\delta)$, ten volts across the top. The output between the two midpoints comes out of solveDC like anything else here:

$$V_{out} = V\left(\frac{R(1+\delta)}{R(2+\delta)} - \frac{1}{2}\right) = \frac{V\delta}{2(2+\delta)}$$

against the expression every reference gives, $V\delta/4$. The ratio of the two is $2/(2+\delta)$, so the linear expression is high by

$$\frac{\delta/2}{1 + \delta/2}$$

— half the fractional change, to first order. Measured on solves at eight values of δ spanning two and a half decades, the departure agrees with that closed form to $10^{-9}$ absolute.

Bisecting for the change at which the tangent is one per cent wrong gives 2.02%, and for a tenth of a per cent, 0.200%. The gate checks that those two stand in the ratio of ten to one, which is the statement that the departure really is first order in δ rather than something with a knee in it.

That is a very small linear range. A resistance thermometer moving 0.39% per kelvin is one per cent out of its tangent after five kelvin. A strain gauge at 1000 microstrain with a gauge factor of two is at δ = 0.2%, which is 0.1% of nonlinearity — small, but the same size as everything else in a good instrument’s error budget, and it is systematic rather than random so no amount of averaging removes it.

Driving it with a current halves it #

Replace the voltage source across the bridge with a current source and solve again. Everything else is identical — the same four resistors, the same output node pair — and the departure at every change is exactly half:

Read the table diagonally and it says something stronger than “half”. The current-driven error at δ is the voltage-driven error at δ/2, at every entry — so the two are the same function with the argument halved, and the one per cent edge moves from 2.02% to 4.04%, exactly double.

The mechanism is that a voltage-driven bridge has a total resistance that changes with δ, so the current through the changing arm falls as that arm’s resistance rises, and the two effects compound. A current-driven bridge holds the total current fixed, so only the division between the two halves changes.

This is a change of one component in the excitation and no change at all to the thing being measured, and it doubles the range over which the standard expression is usable. It is not free — a current source with the stability and noise of a voltage reference is a harder part — but the trade is between two things a designer chooses, neither of which is the sensor.

Two arms remove it exactly #

The nonlinearity measured above belongs to the arrangement rather than to bridges. Change two arms in opposite directions — one gauge in tension and one in compression, which is what any bending member provides for free — and the second-order term does not merely shrink. It vanishes.

Those are not small numbers; they are the floor of double-precision arithmetic, which is the same distinction the semiconductors field draws about a differential pair’s even harmonics. The half bridge’s output is $V\delta/2$ and the full bridge’s is $V\delta$, both exactly, at a twenty per cent change as readily as at a tenth of a per cent.

The reason is a cancellation in the algebra that the solve confirms rather than assumes. With $R(1+\delta)$ over $R(1-\delta)$ the denominator of the divider is $2R$ — the δ terms cancel — so the output is a linear function of δ with no denominator left to expand.

And the sensitivity rises with it. The full bridge’s output is four times the tangent — $V\delta$ against $V\delta/4$ — and therefore $4(1+\delta/2)$ times the quarter bridge’s actual output, since that one is itself low by the factor this essay opened with. At δ = 1% it is 100.0 mV against 24.88 mV, a ratio of 4.020, and the gate holds the exact expression at four changes rather than the round number, which fails at the third digit for exactly the reason the essay is about.

So the correct summary of this essay’s first result is narrower than it first appears, and much more useful for it. A quarter bridge is nonlinear by half the fractional change; a half or full bridge is not nonlinear at all. The choice between them is usually made on grounds of sensitivity and of temperature cancellation, and the linearity comes along with it unremarked — which is a pity, because it is the larger effect for anything measuring a big change.

The leads, which are the other half of every bridge #

A gauge is rarely next to its instrument. Putting resistance in the two leads that carry the excitation into the changing arm changes the answer, and by a proportion that is easy to state:

A tenth of a per cent of the gauge’s own resistance in each lead costs a tenth of a per cent of the reading, which is the intuitive answer and worth confirming rather than assuming. Three and a half ohms — ten metres of thin copper, there and back — costs one per cent, which is five times the nonlinearity this essay opened with.

And the lead resistance is not stable: copper’s temperature coefficient is 0.39% per kelvin, so ten metres of lead swinging twenty kelvin moves the reading by 0.08% all by itself, in a way that looks exactly like the quantity being measured.

The instruments field has already measured this problem and its standard repair.

The last setting is the one worth reaching, because it is where the two arrangements stop being variations on each other. Three and a half ohms is a long lead of thin wire, or a short one of printed copper, and it is not a pathological number.

Both of those figures are the general case of what the table above measures. Two terminals measure the leads as well puts fifty milliohms in each lead of an ordinary resistance measurement and finds the reading one per cent high at ten ohms, ten per cent high at one and a hundred per cent high at a tenth — the reading being the resistance plus the wire, with no approximation in it. A bridge is better off than that by the ratio of its arm resistance to the lead, which is why 350 Ω is a common arm value and 120 Ω is a harder one, and it is better off in the same way for the same reason rather than by any different mechanism.

The instrument at the other end is subject to the identical argument in the other variable. The probe is part of the circuit measures an oscilloscope probe on a two-kilohm source as one per cent wrong at 6.8 kHz — not because the instrument is inaccurate but because its hundred and fifteen picofarads are across the node. A bridge output is a lower impedance than that and a strain measurement is a much lower frequency, so the same product lands harmlessly here; what does not land harmlessly is the direct-current half of it, which is the divider, and the thing it does not know about applied twice over. Two dividers of identical ratio give six volts and one volt into the same load in that essay, and the reason a bridge escapes it is that both of its dividers are loaded by the same amplifier input, so what does not cancel is the difference between the two loadings rather than the loading.

The excitation is an amplitude boundary #

A bridge’s sensitivity is proportional to its excitation, so the temptation is to raise it. What stops that is that the bridge dissipates the excitation, and the four resistors being measured are the things doing the dissipating.

At ten volts across a 350 Ω bridge each arm carries 14.3 mA and dissipates 71.4 mW, computed on the same solve as everything else. Where that goes depends on what the gauge is bonded to: a foil gauge on a thick steel member sinks it easily and one on a thin plastic specimen does not.

The rest of this paragraph is a stated model rather than a measurement, and is flagged as such because this collection’s habit is that a number without a solve behind it says so. With a thermal resistance of 100 K/W to ambient, 71.4 mW is a 7.1 K rise; with a residual temperature coefficient of one part per million per kelvin — which is what a self-temperature-compensated gauge achieves on the material it was matched to — that is 7.1 parts per million of apparent resistance change. Against a 1000 microstrain signal at δ = 0.2%, it is 0.36% of the reading: three times the nonlinearity measured above, from the excitation alone.

The two boundaries move in opposite directions with the excitation, which makes them a pair like the regulator’s capacitor two essays back. The nonlinearity does not depend on the excitation at all — it is a fractional error, and both the signal and the departure scale together — while the self-heating error grows as the square of it. So raising the excitation buys signal against the noise floor and costs accuracy against the thing being measured, and the crossover for the constants above is at about five volts.

The offset that is there before anything moves #

Every table above starts from a balanced bridge, and no bridge is balanced. Four resistors of a stated tolerance are four independent numbers, and the output with δ = 0 is whatever their mismatch makes it — an offset that arrives before the sensor has done anything and is the same size as a large signal.

The tolerance that is not on any part is the measurement of exactly that arithmetic, on exactly four resistors in exactly a divider. Its results transfer without modification: the worst case is the tolerance itself, the measured spread over a population is 0.29 of a per cent for one per cent parts, and the root-sum-square bound offered everywhere as though it were a standard deviation is 1.70 of one. So a bridge of one per cent resistors has an initial offset that can be a full per cent of the excitation and typically is not, and the gap between those two statements is the whole of what a tolerance means. Against the quarter-bridge signal in the table above — 24.9 mV at δ = 1%, on a ten-volt excitation — a one per cent offset is 100 mV, four times the largest signal in this essay.

That offset is a constant and is therefore removable, which is why the arrangement survives at all. It is removable by measurement rather than by design: read the output at rest, subtract it, and what is left is the change. What is not removable is its drift, which is the mismatch of four temperature coefficients rather than of four resistances, and which is what the closing section of this essay is about.

The distinction matters because the two errors this essay has measured are of the third kind again. The nonlinearity is not an offset and not a drift; it is a systematic function of the signal, present in every reading, identical in every unit, and invisible to a zeroing procedure — which is what makes half of a fractional change worth a figure while a per cent of initial imbalance gets a sentence. What a network answers, and how the answer is checked is the routine underneath all of it: one matrix, one answer, checked twice, with the tolerance applied to the netlist rather than to the result.

What is left, and where it is measured #

Three things this essay does not cover, each of which the collection has already measured elsewhere in another guise.

Temperature. Every resistor in the bridge has a temperature coefficient, and the arrangement cancels those that are common to all four. What survives is the mismatch between them, which is the same argument the semiconductors field makes about a differential pair.

Amplification. The output is a differential voltage on a common-mode pedestal of half the excitation, so the amplifier that reads it needs common-mode rejection — which is the same quantity the previous rung measured as rail rejection, in a different place in the circuit. The instruments field has the numbers. The rejection four resistors decide finds that a difference amplifier built from tenth-per-cent parts rejects by 53.99 decibels against a closed form’s 53.98, and that the amplifier has nothing to do with it: the figure is one plus the gain over four times the resistor tolerance, so it is four resistors deciding what happens to a bridge’s five-volt pedestal. The four resistors that decide, and the two that do not puts a two-amplifier stage in front and gains exactly twenty times the log of its gain — because the input stage passes common mode at unity and the improvement is entirely the differential signal arriving larger, which is precisely the arrangement a bridge wants.

And there is a corner in it that belongs to the bridge rather than to the instrument. The corner the instrument has no part in connects a 95 dB instrument to a source with a kilohm of imbalance and ten picofarads at each input, and the rejection falls twenty decibels a decade above 290 hertz, reaching 84 dB at a kilohertz on a part that is still doing 95. What converts common mode to differential there is the difference of two time constants, and a bridge supplies exactly that: two output arms whose resistances differ by δ, which is the signal, feeding two inputs whose capacitances differ by whatever the wiring makes them. The imbalance is not a defect in the bridge. It is the measurement.

Nonlinearity of the sensor itself. Everything here is about the circuit’s nonlinearity, with a resistance that is assumed to move in exact proportion to whatever is being measured. A strain gauge’s own gauge factor varies with strain, a thermistor’s law is exponential rather than linear, and those are much larger effects than the half-a-delta measured here. Separating them is exactly why this essay measures the circuit alone.

The gate #

The output is below the tangent by half the fractional change, at eight changes spanning two and a half decades, agreeing with the closed form to $10^{-9}$.

A current-driven bridge doubles the change it stays inside one per cent for, 4.04% against 2.02%, to better than a per cent of the factor of two.

The edge moves by a decade for a decade of tolerance — 0.200% at a tenth of a per cent against 2.02% at one per cent — which is what makes the departure first order rather than merely small.

A tenth of a per cent of lead resistance in each arm is visible in the reading, which is the claim that the leads belong in the netlist rather than in a footnote.

A half bridge and a full bridge are linear to the arithmetic’s floor, at every change out to twenty per cent, and the full bridge is four times as sensitive as well.

Read together those five say something about the whole field. Four of them are boundaries of the usual kind — a model good over a range, with the range measured — and the fifth is a boundary that a change of arrangement removes entirely. The collection has met that before, in the differential pair whose even harmonics are absent rather than small, and it is the most useful kind of result there is: not a number to design around, but a symmetry that makes the number stop existing.

What survives the symmetry is everything else in this essay. The leads still measure themselves, the excitation still heats the gauges, the amplifier still loads the output, and the sensor still has a law of its own that no arrangement of resistors linearises. A bridge is four resistors, three wires and a number, and the number belongs to whichever of them was not cancelled.