The voltage four wires carry in

Two terminals measure the leads as well put a number on why four wires are used: fifty milliohms in each lead makes a two-wire reading a hundred per cent high at a tenth of an ohm. The voltmeter four wires do not remove found what four wires leave behind at the other end of the range, the voltmeter’s own input resistance. Both treat the sense loop as what the method says it is: a pair of wires that carries no current, so has no drop, so reports the voltage across the device exactly.

The loop carries no current, and it is not free of voltage. Every place a copper lead meets a terminal of another metal is a thermocouple, and if the two such junctions in the loop sit at different temperatures their thermoelectric voltages do not cancel. A microvolt is a modest thermal EMF — a degree of difference between two copper-to-brass joints — and it is the whole of the signal when a milliohm carries a milliamp. Four wires were built to keep current out of the sense loop. The voltage that matters at low resistance comes in by exactly that loop, and the method has nothing to say about it. What removes it is reversing the current, and the question this essay measures is how well, when the EMF does not hold still.

The EMF, and why four wires do not touch it #

The measurement is a netlist: a current source forcing a test current through twenty milliohms of lead, a one-milliohm device and twenty more, and a sense pair of twenty-milliohm leads from the device’s terminals to a ten-megohm voltmeter. A voltage source in series with one sense lead is the thermoelectric EMF. The reading is the meter’s voltage over the forced current, solved at each polarity.

Ten milliamps through a milliohm is ten microvolts, so a microvolt of EMF is a tenth of the reading and the single-polarity reading runs from 0.70 to 1.30 of the truth across three microvolts either way. The netlist agrees with the closed form — the drop plus the EMF, over the current, less the meter’s loading of two twenty-milliohm leads into ten megohms — to a part in a billion. At the opposite polarity the drop changes sign and the EMF does not, so the mean of the two readings, each taken as voltage over current, is the resistance with the EMF gone, at every EMF, to the 4.1 parts in a billion the meter’s loading costs.

That is the reversal in its ideal form: an exact cancellation of anything in the sense loop that does not depend on the current’s direction. It needs no knowledge of the EMF, its size or its sign. It needs the EMF to be the same at both readings, and that is the assumption the rest of this essay tests.

An EMF that is still settling #

Thermoelectric voltages move whenever temperatures do. A hand on a terminal, a draught, a connector just tightened, the shunt’s own heating spreading into its terminals — each sets a junction’s temperature relaxing toward a new value over seconds to minutes. The EMF here is a settled half-microvolt plus a microvolt relaxing with a time constant of a minute, the shape of a junction recovering from being touched.

A single polarity carries the whole EMF whether it is moving or not: 1.35 per cent at ten seconds and still 0.51 per cent five minutes later, when the junction has settled. A settled EMF is no better than a moving one if nothing removes it. The reversal removes everything that is the same at its two readings, and what is left is the change between them, halved: 70.0 ppm at ten seconds, falling to 0.557 ppm as the relaxation dies away. The closed form is exact for an exponential — the residual is the relaxing part times $(1 - e^{-\Delta t/\tau})/2$ over the drop — and the figure holds it at three instants to a part in a million.

The + − + delta goes one step further. Its three readings, weighted a quarter, minus a half and a quarter, cancel a constant and also a straight-line drift, leaving the second difference of the EMF: 0.578 ppm at ten seconds, 0.0046 at three hundred. Four readings, + − + −, weighted one, three, three and one over eight, cancel a quadratic as well and leave the third difference, 0.0048 ppm at ten seconds — a hundred and twenty times better than the delta at the moment the junction is moving fastest.

Each reading added is a derivative removed #

The pattern has a clean statement. Each scheme is a finite difference of the readings with the polarity alternating, and a finite difference of order $n - 1$ annihilates a polynomial of degree $n - 2$ in the EMF. What survives is the drift’s $(n - 1)$-th derivative times the reading interval to that power.

The slopes are one, two and three to three figures: halving the interval halves the reversal’s residual, quarters the delta’s and divides the four-reading scheme’s by eight. That turns a question about a particular junction into a design rule about the measurement. A reading interval short compared with the thermal time constant is worth a power of that ratio per reading added, and an instrument that can switch its current quickly enough to take its readings in a hundredth of the time constant gets a factor of a hundred from each extra reading.

It is the same arithmetic as the numerical integration one step, computed twice checked a step response with: a method’s error goes as a power of its step, and the power is set by how many terms of a Taylor series it cancels. Here the “step” is the interval between readings and the series is the EMF’s drift; the delta is a second-order method and the four-reading scheme a third-order one.

Against the shunt’s own heating #

A larger test current makes the EMF a smaller fraction of the drop — every EMF error in the hero figure falls as one over the current — and it heats the shunt. A manganin shunt of 20 ppm per kelvin and 20 kelvin per watt dissipates $I^2R$ and rises by $I^2RR_{th}$, so its resistance is high by $\alpha R_{th} I^2 R$: 0.400 ppm at an ampere, forty at ten. The resistance that depends on the reading measured that heating as the fourth of a shunt’s errors and the only one that depends on the current. Here it meets the EMF from the other side.

The total, $A/I + BI^2$ with $A$ the scheme’s residual EMF over the resistance and $B = \alpha R_{th} R$, is least at $I^* = (A/2B)^{1/3}$. For one polarity that is 11 amperes, where the reading is out by 149 ppm in all; reversed, 1.8 amperes and 4.08 ppm; with the delta, 0.37 amperes and 0.167 ppm. A better reversal scheme leaves less EMF to beat and so wants less current — the reverse of the usual instinct that a more careful measurement uses more signal.

The least error is $\tfrac{3}{2}A^{2/3}(2B)^{1/3}$, and its dependence on the shunt is a cube root: a shunt with ten times the temperature coefficient costs only 2.15 times the error, because the best current falls to meet it. Its dependence on the reading scheme is the two-thirds power of the residual, and the schemes’ residuals differ by orders of magnitude. So the reading method moves the error by a factor of nine hundred between one polarity and the delta at 20 ppm per kelvin, and the choice of alloy moves it by a cube root. A designer with a budget to spend on a milliohm measurement should spend it on switching the current, not on the shunt.

A worked measurement: a busbar joint #

A bolted joint between two copper busbars is a resistance of the order of a hundred microohms, and it is measured four-terminal precisely because the leads would swamp it. With an ampere through it the drop is a hundred microvolts, ten times the hero figure’s signal at ten milliamps through a milliohm, so every error in the figures is ten times smaller in proportion. Thirty seconds after the clips are attached — the hand that attached them is the disturbance — one polarity is out by 1.1 per cent, a reversal a second apart by fifty parts per million, and the delta by four tenths of one.

The joint is also where the method meets its least favourable junctions. The sense clips grip copper that has been carrying load current and is warm, the clips themselves are plated steel or brass, and the two clips sit at different distances from whatever is heating the bar. An EMF of a few microvolts is ordinary, and it does not settle while the bar’s temperature is still moving. A reversal is the minimum; a joint inspected while its circuit is live needs the delta, and the reading interval short against how fast the bar is warming. The ammeter that is a resistor found a shunt’s resistance chosen by a burden optimum, and the millivolts in the wire the voltages a shared return puts into a measurement; a joint test has both, and the thermoelectric part is the one only the polarity can separate.

Every loop that carries no current #

The four-terminal method is one instance of a habit this subject uses repeatedly: make a loop that carries no current, so that its impedance does not matter, and read a voltage across it. A guard does the same — the sign of what the guard gives back found a guard driven with a copy of the voltage on the node it protects, so that almost no current crosses between them — and so does a high-impedance probe, which the probe is part of the circuit found to be a capacitance rather than nothing. In each case the trick removes the impedance of the loop from the answer and leaves every source in the loop exactly where it was. A loop with no current has no drop and is not therefore a loop with no voltage.

The thermoelectric EMF is the purest example, because it is a source rather than a loading: it does not depend on anything the measurement does, so no arrangement of the loop’s impedances reaches it, and the only handle on it is the one the reversal uses — that it does not change sign when the current does.

How a meter does it #

Precision ohmmeters and nanovoltmeters offer exactly these schemes, under names like offset compensation and delta mode, and the figures say how to read their specifications. An offset-compensated reading is a reversal — or, in some instruments, a reading with the current on and one with it off, which removes the EMF the same way and costs half the signal. A delta-mode reading is the + − + triple, run continuously, each new reading combined with the two before it, so it cancels a linear drift at every point of a long measurement.

Both rely on switching the current quickly against the thermal time constants of the junctions, and both are limited by what the switching itself does. A reversal that heats the shunt differently on each polarity — because the shunt’s terminals have different thermal paths — makes an EMF that is correlated with the polarity, and no finite difference of alternating readings removes a voltage that alternates with them. That is the one EMF the schemes cannot see, and it is why the best shunts are built symmetrically.

Where the EMF comes from #

The junctions are wherever two metals meet in the sense loop: the copper sense lead and the brass or nickel-plated terminal, a solder joint, a relay contact inside the meter, the device’s own terminal and its resistive element. Each pair has its own coefficient, from a fraction of a microvolt a kelvin for copper against copper alloys to tens for some contact platings, and each EMF is the coefficient times the temperature difference between the two junctions of the pair. The settling time constants are those of the terminals’ thermal masses — seconds for a lead clip, minutes for a heavy shunt’s terminal block.

The pulse that ends before the heat found a shunt’s self-heating reaching its terminals through a ladder network of thermal time constants, and those terminals are the junctions of the sense loop. A current that heats the shunt also warms its terminals, unequally if the shunt is unequally mounted, and makes an EMF that appears when the current starts and drifts as the heat spreads. The disturbance in the figures is generic; in a real measurement the test current itself is often the disturbance.

What the measurement leaves out #

The EMF is one lumped source with one exponential relaxation. A real loop has several junctions with several time constants, and the residual each scheme leaves is the sum of their finite differences; the slowest dominates a reversal’s residual and the fastest a delta’s. The current switches instantly; a real source settles, and readings taken before it has settled carry its transient, which a reversal treats as a polarity-correlated error. And the shunt’s heating is its steady state at each current; a measurement short against the shunt’s own thermal time constant sees less of it, which moves the best current up, by the amount the pulse essay measured.

The voltmeter is ideal apart from its input resistance. A real one has its own offset and its own thermoelectric junctions at its input terminals and inside its input relays, and they sit in the same loop as the external ones and are removed by the same reversal — which is why an instrument that offers delta mode can quote a resolution far below its own offset specification. What it cannot remove is its own noise, which the reversal’s finite differences add in quadrature: the delta’s weights of a quarter, a half and a quarter make its noise $\sqrt{6}/4$ of one reading’s, and the four-reading scheme’s $\sqrt{20}/8$.

The number to keep #

A thermoelectric EMF sits in the sense loop four wires keep free of current, so four wires do not remove it: a microvolt is a tenth of a milliohm’s reading at ten milliamps. Reversing the current removes any EMF that holds still, leaving the drift over one interval; the + − + delta leaves the second difference, four readings the third — residuals that fall as the first, second and third power of the reading interval. Against the shunt’s heating each scheme has a best current, $(A/2B)^{1/3}$: 1.8 A and 4.08 ppm reversed, 0.37 A and 0.167 ppm with the delta, for a 20 ppm/K shunt. The least error goes as the cube root of the shunt’s coefficient and the two-thirds power of the scheme’s residual.

Still open: the EMF that alternates, the junction that is the device, and the alternating current #

An EMF correlated with the polarity. A shunt whose two terminals warm differently on each polarity makes an EMF that follows the current’s sign, and no alternating scheme removes it. How large it is for a stated asymmetry in the terminals’ thermal paths would put a number on why a symmetric shunt matters.

The junction that is the device. A thermocouple’s own element, a Peltier junction, or a resistor whose element and terminations are different metals carries an EMF driven by the measurement’s own heating, proportional to the current rather than to its square. That EMF reverses with the current and looks like resistance. Separating it from the true resistance is a different measurement, and it needs a second current.

Measuring with alternating current. A sinusoidal test current at a few hertz is a continuous reversal, and a lock-in detector reads only the part of the voltage at the drive frequency. Its rejection of a drifting EMF is the frequency response of the detector at the drift’s frequencies, and comparing it with the delta’s finite differences would say when a lock-in is worth its complexity.