Two terminals measure the leads as well

An ohmmeter forces a current through something and measures the voltage that appears. If it forces the current down the same pair of wires it measures across, it measures the wires too.

That sentence is the whole of the subject, and everything interesting about it is in how large the consequence is at what resistance.

Two netlists, not two formulas #

Both arrangements here are built as networks and solved, and the reason is that the four-wire case has a subtlety that a formula would hide.

The two-wire netlist is a current source, a lead, the resistance under test, a second lead, and a voltmeter across the point where the current enters and leaves. The voltmeter is a ten-megohm resistor, because a voltmeter is a resistor.

The four-wire netlist forces the current down the same pair, and adds a second pair tapping the resistance’s own terminals, with the voltmeter across those. The sense leads have the same fifty milliohms as the force leads — they are the same kind of wire — and the point is that it does not matter, because almost no current flows in them.

“Almost no” is the subtlety. The sense pair is not carrying zero current; it is carrying whatever ten megohms takes at the voltage being measured, and the sense leads’ resistance appears in series with that. The four-wire error is therefore not zero either. It is the ratio of the sense lead resistance to the voltmeter’s input resistance, which for fifty milliohms and ten megohms is 5×10⁻⁹ — and the measured four-wire error across the whole range is between 10⁻⁶ and 10⁻³ per cent, which is that, plus the divider the voltmeter forms with the resistance under test.

A formula for the four-wire case would have said “zero”. The netlist says what it actually is, and what it actually is tells a reader which instrument specification decides the answer.

The numbers, and where the crossing is #

Two-wire, with fifty milliohms per lead: 0.099% at a hundred ohms, 1.000% at ten, 10.000% at one, 100.000% at a tenth, 1,000% at a hundredth.

The relation is exact and trivially so — the reading is $R + 2R_\mathrm{lead}$, so the error is $2R_\mathrm{lead}/R$ — and the one-per-cent crossing is at $2R_\mathrm{lead}/0.01$, which is 10 Ω. The figure asserts that: at exactly ten ohms the solved two-wire network gives 10.09999 Ω, which is 1.000% high to a part in a thousand of the claim.

The residual is worth a word, because it is not rounding. The solved network also contains the voltmeter’s ten megohms in parallel with everything, and a milliohm of return resistance, so the solved answer is not exactly $R + 2R_\mathrm{lead}$. It differs from it by about a part in ten thousand, which is the size of those two effects, and asserting the simple relation to a part in a thousand is therefore asserting the dominant term while leaving room for the ones that are genuinely present.

Why fifty milliohms is a realistic number #

The lead resistance is the parameter the whole page turns on, so it is worth saying where fifty milliohms comes from rather than treating it as an arbitrary choice.

A metre of ordinary test lead is a few tens of milliohms: copper’s resistivity is 1.7×10⁻⁸ ohm metres, so a metre of wire a millimetre in diameter is about 22 mΩ. Two of those, plus the resistance of the two banana plugs at each end and the two crocodile clips, comfortably reaches fifty milliohms per lead and often exceeds it.

The connections are the larger and less predictable half. A clean plug-and-socket pair is a few milliohms; a tarnished one is tens; a crocodile clip on an oxidised surface can be hundreds and can change while the measurement is being taken. That variability is the second reason the two-wire arrangement is unsatisfactory at low resistance: not only is the error large, it is not repeatable, so it cannot be subtracted.

Subtracting it is what a two-wire instrument’s “relative” or “null” function does — short the leads together, record the reading, and subtract it from everything afterwards. That works, and it works exactly as well as the assumption that the lead resistance has not changed since the null was taken. Flexing a lead changes it. Warming it changes it: copper’s temperature coefficient is about 0.4% per kelvin, so a lead that warms ten degrees in a rack changes by four per cent of fifty milliohms, which is 2 mΩ — an error the null cannot remove and which is a fifth of a per cent at one ohm.

What the four-wire arrangement is actually doing #

It is worth being precise, because the usual summary — “it eliminates lead resistance” — is right about the outcome and unhelpful about the mechanism.

What it does is separate the two functions. The force pair’s job is to establish a known current, and its resistance affects only how much voltage the source has to produce, which the source can handle. The sense pair’s job is to report a voltage, and its resistance affects only the current the voltmeter draws, which is nearly zero. Neither lead resistance appears in the answer, because neither is in a path where it would.

Two consequences follow that the summary does not give.

Where the sense pair connects is the whole measurement. The four-wire arrangement measures the resistance between the two points where the sense leads touch. Connect them a centimetre away along the same conductor and that centimetre is included. This is why a four-terminal shunt resistor is a manufactured component with four terminals on it rather than a resistor with two extra wires soldered on — the sense connections have to be at defined points, and the definition of what is being measured is where they are.

The force current has to be large enough to measure and small enough not to heat. The voltage across a milliohm at one milliampere is a microvolt, which is at the edge of what an instrument can resolve against its own noise; raising the current to an ampere gives a millivolt, which is comfortable, and dissipates a milliwatt in the component, which may not be. That trade is the noise field’s argument arriving in a measurement, and it is where the lower end of this figure’s axis actually stops in practice.

The same problem in the source #

There is a mirror image of this argument that this collection has already made, and putting the two side by side is worth doing.

The source that is not a source is about a voltage source with resistance in series with it: what it delivers depends on what is drawn from it, and the number on the label is what it produces into nothing at all. That is the supply side of exactly this problem. A four-terminal measurement is what a sense connection does about it, and a regulated supply with remote sensing is the same technique applied in the other direction — the regulator’s feedback is taken at the load rather than at its own terminals, so the drop in the supply leads is inside the loop and corrected rather than outside it and delivered.

The two are the same circuit with the roles exchanged, and neither is available without a second pair of wires. That is the honest cost of the technique: four conductors instead of two, and a component or a connector designed for it.

Where the two-wire measurement is fine #

The figure is drawn to make two-wire measurement look bad, and it is worth being fair about where it is not.

Above about a kilohm with ordinary leads, the two-wire error is under a hundredth of a per cent and is smaller than the instrument’s own accuracy. Almost every resistance anybody measures is in that range, which is why two-wire measurement is the default and why most people never think about this at all.

The trouble starts at the low end, and the low end is where the interesting components are: current shunts, contact resistances, cable resistances, winding resistances, the resistance of a connector, the resistance of a solder joint. Those are milliohms to ohms, and every one of them is in the region where a two-terminal reading is mostly leads.

The crossing point is the useful number and it scales exactly: with 5 mΩ leads it is 1 Ω, with 50 mΩ it is 10 Ω, with 0.5 Ω it is 100 Ω, with 2 Ω it is 400 Ω. Multiply the lead resistance by two hundred and that is where a two-wire reading is one per cent high.

That factor of two hundred is worth carrying around, because it converts a quantity nobody measures into a rule anybody can apply. Estimate the lead resistance — a few tens of milliohms for ordinary leads, a few hundred for a clip on a corroded surface — multiply by two hundred, and any resistance below the answer is being measured mostly through the leads. Everything above it is fine to a per cent, and the transition is a smooth inverse rather than a threshold, so a factor of ten below the crossing is ten per cent and a factor of ten above is a tenth of a per cent.

There is no ambiguity anywhere in that, which is unusual for a rule of thumb. It follows from $R + 2R_\mathrm{lead}$ being the exact reading of the two-wire arrangement, and the only estimate in it is the lead resistance itself.

What the field adds up to #

Three instruments, three mechanisms, one argument.

A probe puts capacitance across a node and its reading is one per cent wrong above a frequency set by that capacitance and the source’s resistance. A compensated divider has two ratios and they agree only inside a four-and-a-half per cent window of one component’s value. An ohmmeter’s leads are in series with the thing being measured and dominate below a resistance set by their own.

None of the three is an inaccuracy in an instrument. Every one of them is the instrument doing exactly what it is built to do, in a circuit that now includes it — and in every case the size of the effect is computable in advance from two numbers that are both on the specification sheet.

The three fixes are worth comparing, because they are structurally different and the differences are instructive. The probe’s fix is attenuation: divide before the capacitance, and pay a factor of ten in signal for a factor of ten in bandwidth. The compensated divider’s fix is cancellation: make a pole and a zero coincide so that the frequency dependence disappears, and pay for it with an adjustment that has to be maintained. The four-terminal arrangement’s fix is separation: move the offending resistance into a path where it does not enter the answer, and pay for it with two more wires.

Only the third of those is free of a residual. An attenuator still loads; a cancellation still drifts; a separated path is simply not in the answer, and its resistance can be anything at all. That is why the four-terminal arrangement is the one of the three that is described as eliminating its error rather than reducing it, and the description is very nearly accurate — the residual it leaves is the voltmeter’s own input resistance, six orders below the effect it removed.

One thing this page does not model #

The two-wire error described here is entirely resistive, and there is a second error at low resistance that is not.

Every junction between two dissimilar metals is a thermocouple, and a measurement circuit contains several: copper to solder, solder to the resistive element, copper to the connector’s plating. Each produces a few microvolts per kelvin of temperature difference, and at low resistance a few microvolts is a large signal — the voltage across a milliohm at a milliampere is one microvolt, so a one-kelvin gradient across a single junction can be several times the quantity being measured.

The standard answer is to reverse the current and average the two readings, since the thermal voltage does not reverse and the resistive one does. That is a genuinely different technique from the four-wire arrangement, it solves a genuinely different problem, and this page does not model it at all: the netlists here contain no sources other than the forcing current and no temperature anywhere.

Naming it is the honest thing to do, because a reader who takes this page as a complete account of low-resistance measurement would have the four-wire technique and none of the thermal one, and at milliohms the second matters as much as the first. The site’s rule asks every model to carry the condition under which it stops applying, and the condition here is a statement about physics that is not in the model rather than a frequency or an amplitude that is.

Where the same arithmetic decides a design rather than a measurement #

The correction this essay is about — a series resistance in a path where somebody was thinking about a different quantity — appears three more times in this collection, and in each of them it is a design parameter rather than an instrument error.

The ammeter that is a resistor is the closest: every direct measurement of a current is a measurement of a voltage across something the current was made to flow through, so the instrument has two errors pointing opposite ways and the best of them sits at the geometric mean — 7.75 millivolts of burden on a twelve-volt rail, with a worst-case error of 0.129 per cent, and neither number containing a resistance or a current. A shunt is this essay’s lead resistance deliberately fitted, sized rather than apologised for.

The bridge that is linear near one point has the same ohms in the same place and turns them into a gain error: three and a half ohms in each excitation lead — ten metres of thin copper, there and back — costs one per cent of the reading on a 350 Ω bridge, which is five times the nonlinearity that essay’s whole argument is about. And because copper moves 0.39 per cent per kelvin, ten metres of lead swinging twenty kelvin moves the reading by 0.08 per cent all by itself, in a way that looks exactly like the quantity being measured.

And the millivolts in the wire is the case where the shared conductor carries somebody else’s current: ten millimetres of one-ounce copper is five milliohms, and a hundred-milliamp load returning through it puts half a millivolt into a ten-millivolt sensor’s reading — five per cent, before anything has been amplified. Four wires fix this essay’s problem because the sense pair carries no current; they do nothing about that one, because the interfering current is in the conductor whether or not the instrument’s own current is.

The four together are the field’s opening argument in four circuits: a wire is an element in the netlist, and which of its properties matters is decided by what else is connected to it.

Which property that is changes with frequency, and the change has a corner that no amount of copper moves. The millivolts in the wire measures ten millimetres of one-ounce copper as five milliohms and ten nanohenries, with the crossover at 79.6 kilohertz — and finds that shortening the run moves the whole curve down and the corner not at all, because both quantities scale with the geometry together. So a four-wire measurement removes a resistance, and the inductance it leaves behind is the term that matters above eighty kilohertz and is not removed by sensing separately.

For a direct-current measurement of a milliohm that is irrelevant, which is why the technique on this page is complete for the job it is used for. It stops being complete the moment the current being forced through the unknown is not steady — a pulsed measurement, a lock-in at a few kilohertz, a milliohm meter that chops to escape thermal voltages — and at that point the sense pair is reading a resistance plus an inductive term that grows with frequency, from a conductor whose corner is a property of copper rather than of the instrument.

The one error this field removes #

A four-terminal connection is the only arrangement in the instruments field that removes an error rather than bounding it, which is why the rest of the field is a list of bounds. The probe is part of the circuit is a bound: the loading is computed and lived with. The millivolts in the wire is a bound: the shared impedance is computed and rejected as far as four resistors allow. The ammeter that is a resistor is where this page’s repair is applied to a shunt, and The divider, and the thing it does not know about is the general form of the error being removed.