The millivolts in the wire

This field’s standing argument is that an instrument is an element: it goes in the netlist, and every reading is a reading of the circuit that includes it. Three essays have applied that to the thing doing the measuring — a probe, a divider, a pair of leads.

This one applies it to the thing that is never drawn at all. Every schematic in the collection has a ground symbol on it, several times, and the symbol asserts that all those points are the same node. On a board they are not. They are the two ends of a conductor, and a conductor is an element with a resistance and an inductance, both proportional to its length.

Put two circuits on that conductor and one of them is now in the other’s netlist.

What is in the netlist that is not on the schematic #

A one-millimetre-wide track in one-ounce copper is about half a milliohm per millimetre and about a nanohenry per millimetre. Both numbers are per unit length and that fact is the whole second half of this essay.

The netlist has five things in it. A five-volt source and a fifty-ohm load, which draw a hundred milliamps. A sensor of ten millivolts, referenced not to the star point but to the local node the load’s return also passes through. A high-impedance amplifier input measuring that sensor against the star point. And between the local node and the star point, the shared conductor: five milliohms in series with ten nanohenries.

The load current has to reach the star point, and the only way there is through the shared conductor. So the local node is not at zero volts. At low frequency it sits at $100\ \mathrm{mA}\times 5\ \mathrm{m}\Omega$ = 500 µV, and the amplifier is measuring the sensor plus that.

Ten millivolts is the signal. Five hundred microvolts is five per cent of it, and it is neither noise nor drift nor an offset: it is a deterministic voltage proportional to a current that has nothing to do with the measurement.

Two routes, and the difference between them #

The site’s habit is to compute a quantity twice by routes that share nothing but the netlist, and this one has an obvious pair.

The whole thing solved at once. Assemble all five elements, solve at a frequency, read the voltage at the local node.

Superposition. Solve a netlist containing only the power circuit and the shared conductor — no sensor, no amplifier — read the current through the conductor, and multiply by $|R_g + j\omega L_g|$.

They agree to $4.4\times10^{-7}$ at ten millimetres, which is close and is not machine precision, and the residue is physical rather than arithmetic. The amplifier’s input divider draws a few nanoamps, and those nanoamps go home through the same conductor — so the second route is missing a contribution that the first one has. The disagreement grows with the shared impedance, reaching $1.5\times10^{-5}$ at a hundred millimetres, which is the essay’s own subject arriving inside its own verification.

The corner, and what it is made of #

The shared conductor’s impedance is flat at $R_g$ and then rises. The corner is at

$$f = \frac{R_g}{2\pi L_g}$$

which is 79.6 kHz here. Below it the error is a fixed 500 µV. Above it the error grows a decade per decade, and there is nothing above it — no second corner, no roll-off, no ceiling. At 1.59 MHz the error is ten millivolts, which is the whole signal.

Two things follow that are worth stating separately.

Gain does not help. The error is in series with the sensor, so it is amplified along with the signal and the ratio is unchanged. Neither is averaging any use: the interference is not random, it is a copy of the load current.

Bandwidth-limiting helps only if the interference is out of band. A load switching at 500 kHz puts its harmonics wherever they land, and a ten-hertz measurement bandwidth removes them. A load whose current has content inside the measurement band — a motor, a heater with a slow controller, a converter’s low-frequency envelope — does not.

Why shortening the run does not move the corner #

The slider is the length of the shared conductor, and what it does is the second finding.

Both $R_g$ and $L_g$ are proportional to length. So their ratio is not, and the corner — $R_g/2\pi L_g$ — is 79.6 kHz at every setting of the slider, from two millimetres to a hundred. The figure asserts that across the whole drag range rather than at the one being drawn: the seven corner frequencies agree to a part in $10^{12}$.

What the length does move is the whole curve, vertically, in proportion.

That is a clean and useful rule. Halving the shared length halves the error at every frequency, and does not change the frequency at which the error’s character changes. There is no length at which the inductance stops mattering and no length at which the resistance takes over again; the two are locked together by geometry.

It is also the reason the repair is topological rather than dimensional. Shortening a shared run by a factor of two is often hard and always buys a factor of two. Not sharing it at all buys everything, because the error is proportional to the shared impedance and a conductor that carries only one circuit’s current has none.

The repair, and what it is called #

Referencing the sensor to the star point directly — its own conductor, carrying only its own current back — removes the term entirely. That is what a single-point ground is, and this essay is what it is for.

The name it goes under is confusing and worth untangling, because “ground loop” describes two different faults that people treat as one. The other one is a genuine loop of conductor that encloses an area and picks up a changing magnetic field — an induced electromotive force, proportional to $dB/dt$ and to the area, and the lines field’s return-path essay is where loop area is measured.

This one has nothing to do with area. It would be exactly the same if the whole circuit were folded flat with no enclosed loop at all, because the mechanism is a shared impedance rather than a shared flux. The two often occur together, they are fixed by different things, and only one of them can be computed from a netlist.

Three places the shared impedance is not obvious #

A four-terminal shunt with three terminals. A current-sense resistor is specified as a four-terminal part precisely to avoid this, and the specification is defeated by taking the sense connection from the pad rather than from the sense tab. A millimetre of pad copper carrying the full load current is a shared impedance of half a milliohm, which on a ten-milliohm shunt is five per cent.

A converter’s reference pin. An analogue-to-digital converter’s reference and its digital return frequently share the last few millimetres to the plane, and the digital return current is a train of spikes with content up to hundreds of megahertz. The error appears as a code-dependent shift in the reference, which reads as integral non-linearity and is not.

A daisy-chained supply. Three boards fed from one supply through one return wire have each other’s currents in their own ground references, in the proportion of how much of the wire they share. The last board in the chain has the most, and moving it to the front of the chain changes the answer — which is a diagnostic, because nothing about a real fault in a board should depend on the order of the wiring.

The model in the conductor has an edge of its own #

Five milliohms and ten nanohenries are constants in the netlist above, and that is a model like every other model in this collection: right below a frequency and wrong above it.

The resistance is the one that goes first. Current crowds towards the surface as the frequency rises, so a conductor’s resistance is its direct-current value only while the skin depth is comfortably more than its thickness. For thirty-five-micrometre copper — one ounce — the skin depth equals the thickness at about 3.5 MHz, and the magnetics field’s own measurement of the same effect shows the resistance already a couple of per cent up where the rule of thumb says the effect begins.

That matters less than it might, and it is worth saying why rather than leaving it as a caveat. Above 79.6 kHz the shared impedance is dominated by the inductive term, which is not affected: at 3.5 MHz the inductive part is 220 mΩ and the resistive part is 5 mΩ, so a resistance that has doubled changes the total by one per cent. The model’s error arrives in the term that has already stopped mattering, which is a piece of luck rather than a design, and is the reason the curve above is trustworthy for four decades past the corner.

The inductance is on firmer ground. The internal inductance of a conductor falls slightly with frequency as the current moves outward, but the external part — which is nearly all of it for a track over a plane — is a geometric quantity and does not move at all.

What this essay does not claim #

That five milliohms per centimetre is a universal number. It is a one-millimetre track in one-ounce copper. A plane is a hundred times better and a wire-wrap link is ten times worse, and the inductance varies less than the resistance does — which makes the corner frequency move, and is the one thing in this essay the slider does not show.

That the amplifier’s input impedance is the fix. It is not, and this is the point of the figure. The error is in series with the source, so it survives any input impedance; the two megohms in the netlist here could be two gigohms and the reading would be the same to five figures.

That the interfering current has to be large. A hundred milliamps was chosen because it makes the arithmetic legible. A milliamp through the same conductor gives five microvolts, which is under a thermocouple’s signal, over a strain gauge’s resolution, and exactly the size of the offsets that get blamed on the amplifier. For scale, the floor a resistor sets puts a kilohm’s thermal noise at 4.00 nanovolts per root hertz, so five microvolts is three orders above the floor of the measurement it is corrupting — and unlike the floor it does not fall with bandwidth, because it is a signal rather than a noise.

That the interfering current has to be a current somebody meant to draw. The shape of it matters as much as the size, and two measurements in this collection produce exactly the wrong shape. The direct voltage that is a sawtooth finds a rectifier drawing its 157 milliamps of average current as a 2.098 ampere pulse over 28.8° of each half cycle — a crest factor of 13.4 that gets worse as the reservoir capacitor is made larger. Through a shared conductor that is not five hundred microvolts of error but thirteen times it, arriving twice per mains cycle. And the first cycle, which no steady state contains finds the first conduction after switch-on at 32.4 amperes against a repetitive peak of 1.23, which through the same conductor is a millivolt-scale event on a microvolt-scale measurement, once, at an instant nobody was recording.

That this is the same as ground-loop pickup. It is not, as above. Both are called ground loops, one is a shared impedance and one is an enclosed area, and a fix for either does nothing for the other.

That subtracting the two ends of the conductor removes it. It removes most of it, and the amount left is a measurement rather than a hope. The rejection four resistors decide takes exactly this circuit — a ten-millivolt sensor measured against a ground somebody else’s hundred milliamps is using — and finds a difference amplifier built from tenth-per-cent resistors removing 53.99 decibels of it against a closed form’s 53.98, which is a factor of five hundred and not a removal. The number is one plus the gain over four times the resistor tolerance, and the amplifier has nothing to do with it. So five hundred microvolts of interference becomes one microvolt, which is back below the sensor’s signal and above its resolution.

And the removal has a corner in it that this essay’s own inductance puts there. The corner the instrument has no part in finds a 95 dB instrument falling twenty decibels a decade above 290 hertz when its source has a kilohm of imbalance and ten picofarads at each input — the mechanism being a difference of two time constants rather than anything about the part. The shared conductor measured here is one arm of exactly that imbalance, so the frequency above which the interference stops being rejected and the frequency above which the interference stops being resistive are both properties of the same piece of copper.

What it looks like when it is found #

The signature is worth writing down because it is unlike anything else that goes wrong with a measurement.

The reading changes when a different circuit changes. Switching the load on and off moves the sensor’s reading by half a millivolt and moves nothing else; nobody touched the sensor, the amplifier or the wiring to either. That is unlike noise, which does not correlate with anything; unlike drift, which correlates with time and temperature; and unlike loading, which correlates with the source impedance of the thing being measured.

It also scales with the interfering current rather than with the signal, so halving the signal doubles the error as a fraction and halving the load current halves it absolutely. Two experiments, each a minute long, and between them they identify the mechanism uniquely.

What the shared impedance costs #

Five milliohms and ten nanohenries of ordinary copper is the error three later measurements in this field are about. The rejection four resistors decide is how much of it survives a difference amplifier, and The rejection the parts have is the ceiling on that rejection. Two terminals measure the leads as well is the one arrangement that removes it rather than rejecting it. The ammeter that is a resistor is where the same copper is the measurement, and Where the current comes back is where its inductance is computed from the geometry.

The gate #

Two routes to the error voltage — the whole netlist solved at once, and the interfering current computed on its own times the shared impedance — asserted to $3\times10^{-5}$ across twenty-four frequencies spanning six decades.

The tolerance is loose by this site’s standards and is loose on purpose. The gap is the sensor’s own return current flowing through the same conductor, which the second route does not contain, and it grows with the shared impedance exactly as the essay’s subject does. Tightening the tolerance would mean deleting the amplifier from the first netlist, which would make the two routes the same computation.

The corner frequency is asserted across the whole slider, not at the setting drawn: seven lengths, seven values of $R_g/2\pi L_g$, agreeing to a part in $10^{12}$. It is the essay’s second finding and it is a claim about the slider rather than about a frame.

And the frequency at which the error equals the signal is bisected on the solved response, so the 1.59 MHz in the caption is a measurement rather than an evaluation of $R_g/2\pi L_g$ times a ratio.

The corner is the part that survives every repair #

The resistance in this conductor can be reduced — a wider track, a plane, a shorter run — and every one of those moves the whole curve down and leaves the corner where it is, because the corner is the ratio of the resistance to the inductance and both scale together with the geometry. That is the essay’s second finding and it is the one with consequences for the rest of the collection, because it means the high-frequency half of this error is not improvable by the usual means.

Two other measurements land on the same conclusion from different directions. Where the current comes back estimates the frequency at which a return current stops spreading and runs directly under its track at 106 kilohertz for any track two hundred micrometres above a half-milliohm plane, with neither the length nor the width in it; and the corner that is three decades wide solves the same plane and finds a band rather than a corner — half gathered by 283 kilohertz and nine tenths by 1.42 megahertz, the length still absent and the width entering only through its ratio to the height. So the geometric quantity that moves a return path is the spacing to the plane, which is a stackup decision rather than a layout one. And the corner the instrument has no part in finds the rejection that is supposed to remove all of this falling twenty decibels a decade above 290 hertz for a source with an ordinary imbalance.

Put together, the three say that a shared-impedance error at direct current is a solved problem — five hundred microvolts becoming one, by subtraction — and that at a hundred kilohertz it is not solved by any of the three techniques, because the interference is rising, the rejection is falling and the conductor’s corner cannot be moved.