Where the current comes back

A schematic has a ground symbol and the current in the return is not drawn. On a board it is drawn anyway, by physics, and the shape it takes is not the same at all frequencies — which means that two circuits that are identical on paper behave differently, and the difference is in the half of the loop the drawing left out.

Two paths, and the cost of each #

A track ten centimetres long, two hundred micrometres wide, two hundred micrometres above a copper plane of half a milliohm per square. The current goes out along the track and comes back through the plane, and the plane offers it a continuum of routes.

Two of them bound the problem. The widest route spreads across as much copper as it can reach, which minimises resistance: 83.3 mΩ here, if its effective width is taken as three times the track’s. The tightest route runs directly beneath the track, which minimises the enclosed loop area and therefore the inductance: 125.7 nH, if the pair is treated as two parallel plates.

Current does not choose. It distributes itself to minimise the total impedance, and which of the two costs dominates depends on frequency: resistance below, reactance above. The crossover is where they are equal,

$$f = \frac{R}{2\pi L}$$

which for this geometry is 106 kHz.

That is an estimate, and it is worth being exact about what kind. Both paths are chosen before anything is solved — a wide one whose width is simply taken as three times the track’s, and a tight one given the inductance of two parallel plates — so the corner is the frequency at which two chosen paths would cost the same, not a frequency at which the current has been seen to move. The corner that is three decades wide solves the plane across its whole width instead and finds no corner: resistance equals reactance at 1.59 kilohertz with the return still where it was at direct current, and the return then gathers beneath the track over three decades, half of the way by 283 kilohertz and nine tenths by 1.42 megahertz. The 106 kilohertz lands about a quarter of the way into that band. The estimate is kept here because the parts of its argument that matter most to a board survive the solve — the length is absent and the stack-up decides — and the sections below say which parts do not.

In the estimate’s picture the return below the corner spreads across the plane and encloses 50 mm² between going and coming back, and above it runs as a strip under the track enclosing 20 mm². The second number survives the solve in a stronger form than the estimate gives it: current flowing in the plane’s own surface and spread symmetrically about the track gives the loop a moment of the track’s length times its height however wide the spread is, so 20 mm² is the radiating area at every frequency, and the 50 mm² belongs to the assumed wide path rather than to anything that radiates.

What the estimated crossover does not depend on #

Sweep the estimate’s geometry and the surprising part is what falls out of the answer.

Length does not appear. Both the resistance and the inductance are proportional to it, so the ratio is not: a quarter of the track has a quarter of each and the same crossover to a digit.

Width does not appear either, for the same reason — a wider track spreads the return over a proportionally wider strip of plane and lowers both quantities together. That one is a property of the estimate rather than of the board, because the wide path’s width was defined as a multiple of the track’s and so cancels by construction. Solved, the track’s width enters the top of the band through its ratio to the height: at the same height, half the gathering falls at 0.91 of the natural frequency below for a track a hundred micrometres wide and at 0.77 of it for one five hundred wide.

What is left is two things:

$$f = \frac{R_\text{sheet}}{2\pi\mu_0 h \cdot s}$$

the sheet resistance of the plane and the height above it, with $s$ the factor by which the low-frequency path is wider than the track. The solve keeps the first two and has no use for the third: $R_\text{sheet}/2\pi\mu_0 h$ is 317 kilohertz at two hundred micrometres, and half the gathering falls at 0.89 of it, a ratio that runs from 0.59 at fifty micrometres to 1.11 at 1.6 millimetres. Everything a designer usually varies — how long a track is, how wide, where it goes — is absent, and the two things that decide the answer are set by the stack-up.

That is a statement worth having, because it converts a question that looks like layout into a question about the board. A four-layer board with a plane 200 µm under the signal layer has its return half gathered by about three hundred kilohertz and nine tenths gathered by a megahertz and a half, on every track it carries; the estimate’s corner for it is a hundred kilohertz. A two-layer board with 1.6 mm of substrate has the same band lower — half by 44 kilohertz and nine tenths by 196 — against an estimated thirteen, and the return under every track on it is gathering through the audio band.

What the tight path is worth #

The reason to care is what encloses the flux, because that is what radiates and what picks up.

At the drawn geometry the tight loop is 20 mm² and the wide one 50 mm². A milliamp going round the tight one at a hundred megahertz produces −1.1 dBµV/m at three metres, which is a long way under any emissions limit and is the reason ordinary boards work.

Move the plane away and it becomes ordinary rather than comfortable:

Six decibels per doubling of height, exactly, because the small-loop field is proportional to the area and the area is proportional to the height. Sixty-four times the height is thirty-six decibels, and thirty-six decibels is the whole of the margin an ordinary design has.

The plane is not there to be a low-resistance ground. Solved, ten centimetres of it is a milliohm while the return is spread and 38 mΩ once the return has gathered under the track, which is where a signal above a couple of megahertz meets it — and as a ground that is a poor one. It is there so that the return has somewhere tight to run, which is a statement about area rather than about resistance — and it is why a plane with a slot cut through it is so much worse than its resistance suggests: the return has to go round, the area grows by whatever the detour encloses, and the resistance barely changes.

The second edge, which the model puts on itself #

The radiated field above comes from the small-loop expression, which is valid while the loop’s perimeter is a small fraction of a wavelength. That is a condition, and it is drawn as a second mark on the figure rather than assumed.

For a 20 mm² loop the perimeter is about 18 mm, so a tenth of a wavelength is reached at about 1.7 GHz. Below that the field rises as the square of the frequency and in proportion to the area, and the expression is a good one. Above it the loop is an antenna with structure and the expression is not an approximation to anything — which is why the mark is drawn at all.

This is the kind of edge this collection is built to make visible, and it once caught the figure lying. The second mark was originally placed on an axis that ended three decades below it, so edgeMark returned nothing and the mark was silently absent at every setting of the slider, under a caption that quoted its value. That is what the site’s record of dropped marks exists for, and it is why the axis now extends to twice the second edge rather than to a round number.

How tight the tight path actually is #

“Directly beneath the track” is the limiting statement and the real distribution is worth having, because it is what decides whether a nearby track shares any of it.

This is the one number in the essay that is quoted rather than computed by the machinery here: for a track at height h over an infinite plane, the return current density across the plane falls off as $1/(1 + (x/h)^2)$, where x is the distance sideways from under the track. Integrating it gives the fraction of the return inside a stated width:

Those fractions are for a track of no width. For this one, two hundred micrometres wide, the full solve at high frequency gives 48.7, 79.4 and 93.7 per cent. Half the current is inside a strip as wide as the track is high. At two hundred micrometres up, that is a strip four hundred micrometres wide carrying half the return, and 97.9% of it inside six millimetres.

Two things follow. The tight path really is tight — the loop area used above, length times height, is the right order and not an idealisation. And the width of the copper a return needs scales with the height above it, which is why a plane close under the signal layer is worth so much more than a wide one further away, and why the answer to a congested layout is usually another layer rather than more copper.

It also says what “sharing a return” means quantitatively. Two tracks whose centres are ten heights apart overlap only in the tails of each other’s distribution; two tracks a height apart are running in each other’s current.

The rule this leaves for a board #

Three sentences, and every number in them came out of the geometry rather than a table.

The band is a property of the stack-up, not of the layout. Its top is set by sheet resistance over height, with a $2\pi\mu_0$ in it and no spreading factor, and the estimate compresses the whole band into one corner. Above the band, every return on the board runs under its own track whatever the designer intended.

Above the crossover the quantity that matters is enclosed area, and area is length times height. Nothing in it responds to making a track wider, and everything in it responds to moving the plane closer or to not making the return take a detour.

And the band is low. Nine tenths gathered by a megahertz and a half on a four-layer board and by two hundred kilohertz on a two-layer one — below the clock of almost everything digital. A designer working above a couple of megahertz is always on the inductive side of it, which means the resistance of a ground plane has almost never been the relevant quantity there. Between ten kilohertz and a megahertz the return is partly spread and partly gathered, and the proportion moves with frequency across the whole of that range.

Where it is met #

A decoupling capacitor’s loop. The capacitor, its vias, the plane and the device pin enclose an area, and above the crossover that area is the whole of what the capacitor’s effectiveness depends on. This is why the vias matter more than the capacitance, and why a datasheet’s equivalent series inductance is the smaller half of the problem.

A signal that crosses a plane split. The return cannot follow, so it goes round — and at anything above the crossover it must, because the tight path is the one it needs. The added loop area is the detour’s, which for a centimetre of split at this height is several hundred square millimetres.

And a measurement made with a long ground lead. A scope probe’s ground clip encloses several hundred square millimetres by itself, so above the crossover the probe is measuring the voltage across its own return inductance in addition to the signal. The instrument field’s version of this is the same statement about the instrument rather than about the board.

The last one is worth a number, because it turns a rule of thumb into an estimate. A six-centimetre ground clip standing a centimetre off the board encloses about 600 mm², thirty times the tight loop this essay computes, so its inductance is of the same order as the whole track’s — a hundred nanohenries or so. A signal with a two-nanosecond edge and fifty milliamps of return current develops $L\,di/dt = 2.5$ V across it, which appears on the screen as ringing and is not on the board at all. That is why a probe’s spring tip exists, and the size of the effect is the ratio of two loop areas rather than anything about the probe’s bandwidth.

What the model contains #

A uniform plane and a track of one width. The effective width of the low-frequency path is taken as three times the track’s, which is not a derivation. Everything that depends on it — the resistance and therefore the crossover — carries that factor. The derivation is the corner that is three decades wide, and of this essay’s structural claims it keeps two, the absent length and the stack-up setting the answer, and removes one, the absent width.

No skin effect in the plane. The sheet resistance is a constant here. In copper the skin depth equals the thickness of a 35 µm plane at about 3.4 MHz, so above there the resistance rises with the square root of frequency — which pushes the crossover further down rather than up, and does nothing to the conclusion.

And no coupling to anything else. The return is treated as belonging to one track. On a real board the plane carries every return at once, they share copper, and two returns overlapping is exactly what common-impedance coupling means.

What the return path decides #

A return current’s path is a geometry, and it decides three quantities the rest of this field uses. A quarter wave, and the path the current takes back is where the loop area becomes a radiated field, and where three boundaries four decades apart belong to the same copper. The receiver that is a branch and The resistor at the wrong end are terminations whose characteristic impedance is set by the same height above the plane. The millivolts in the wire is the instruments field’s version, where the shared part of that path becomes an error voltage, and The staircase in time is the wave picture the impedance is fed into.

The gate #

The crossover is asserted to be a real frequency computed from the geometry, between a kilohertz and ten megahertz, rather than a number the figure carries.

The wide path is asserted to enclose more area than the tight one, which is the statement that there is a trade at all — if it did not, there would be no crossover and no essay.

The small-loop expression is required to refuse itself above its own validity edge. It is fed a frequency three times that edge and must report itself invalid, which is the assertion that has to keep rejecting for the radiated numbers to mean anything.

And every mark the figure asks for is required to have been drawn. The site records every mark a generator requested and an axis could not carry, and the gate refuses a non-empty list — which is the machinery this figure’s own missing mark caused to be written.

What the return path decides for the rest of the collection #

A return current that runs directly under its track once it has gathered — above a megahertz and a half on this stack-up — is a statement about a loop area, and a loop area is the quantity three other measurements here turn out to be functions of.

The millivolts in the wire is the direct-current end of the same argument and the one with the numbers: ten millimetres of one-ounce copper is five milliohms and ten nanohenries, so a hundred-milliamp load and a ten-millivolt sensor sharing it put half a millivolt of somebody else’s current into the reading, and above 79.6 kilohertz that error rises a decade per decade with no ceiling. The two corners are close and they are not the same corner: this essay’s is where the return path changes shape, and that one is where the shared impedance stops being resistive.

The far end that cancels is what the tightened loop does to a neighbour. Two parallel tracks couple through a mutual capacitance injecting a current and a mutual inductance injecting a voltage, and the two add at the near end and subtract at the far end with the same constant in front of both — so when the couplings are equal the far-end crosstalk is $3.5\times10^{-19}$ of the drive, which is zero to the last bits of a double. Equal couplings is what a track over a plane with its return directly beneath produces, so the arrangement measured here is the one that makes that cancellation available.

And the millimetre that becomes common mode is the failure of the same symmetry in the other geometry: a pair whose halves differ in length by a millimetre and a half converts a tenth of its differential signal into common mode by three gigahertz, and the differential signal has lost five thousand parts per million of itself getting there — which is why a pair can pass its own eye and fail an emissions test.

The three together are the field’s practical content. Where the return goes decides the loop area, the loop area decides the coupling to everything else, and none of the three quantities is on the schematic.

Why the answer contains none of the usual variables #

The most useful thing about the 106 kilohertz is what is absent from it. It does not depend on the length of the track or on the current it carries — only on the height above the plane and the plane’s own sheet resistance — and that survives the solve, which is worth dwelling on, because it inverts how a layout decision is usually reasoned about. The width is absent from the estimate and present in the band, but only through its ratio to the height, which is a stack-up quantity too.

A designer worried about a return path reaches first for the things a layout tool exposes: make the track shorter, make it wider, keep it away from other tracks. The first moves nothing, and the second moves the band by a fraction of what the height does. What moves it is the stack-up — the dielectric thickness between the signal layer and its plane — which is chosen once for the whole board, usually by somebody else, and usually for reasons of impedance or cost rather than of return current.

That puts the corner in the same class as the edges that are lengths collects: boundaries nobody chooses at the schematic, set by whoever builds the thing, appearing in no netlist. And it puts it in the same class as Kirchhoff’s own frequency, which is set by nothing but the physical size of the circuit — with the useful difference that the whole band sits below it: nine tenths gathered by 1.42 megahertz, against 3.97 for the same ten centimetres of copper. An ordinary board is above this band at most frequencies a digital signal contains and below the lumped-element boundary at most of them, which is exactly the regime in which the current law still holds and the return path is already tight.

Which is the good news buried in the measurement. Above a megahertz or two the return arranges itself under the track without being asked, so the loop area is small by default and the couplings that depend on it are small by default. What a designer has to avoid is not achieving that arrangement but breaking it — a slot in the plane, a change of layer, a connector — each of which forces the return to detour and enlarges an area that was otherwise minimal without anybody’s help.