Two returns in one plane

The millivolts in the wire measured the oldest problem in layout on a single conductor. A hundred-milliamp load and a ten-millivolt sensor both return through ten millimetres of copper; the copper has five milliohms and ten nanohenries; and half a millivolt of the load’s current appears in the sensor’s reading. The cure everyone learns is a ground plane: give every return a whole sheet of copper, and the returns stop sharing a wire.

They do not stop sharing the plane, and the question is how much of it they share. The corner that is three decades wide solved one track’s return across a plane cut into strips and found that it gathers under the track over three decades of frequency — spread across the whole plane at direct current, concentrated within a height or two of the track far above the band. Where the plane runs out found what happens to that return near an edge. Both essays left the second track out, and both named it as the next thing to solve: below the band two returns overlap however far apart their tracks are routed, and above it they should not.

This essay puts the second track in. One track carries a current; the other carries none; and the figures measure the voltage the first track’s return puts along the second track’s loop. That voltage, per ampere and per metre, is the transfer impedance between the two loops. Its real part is the resistance they share and its imaginary part, over the angular frequency, is their mutual inductance. Together they are the common impedance of the single-wire essay, turned into a property of a plane with two positions on it.

The cross-section with two tracks in it #

The geometry is the one the single-track essays used, so that every number can be set beside theirs: a plane fifty millimetres wide of half a milliohm a square, and tracks two hundred micrometres wide and two hundred above it. The second track sits a stated distance from the first, at the same height. The plane is cut into about 240 strips, graded finely under both tracks and coarsely at the plane’s edges, and at each frequency the strips’ currents are solved so that every strip sees the same voltage drop along its length — which is what a plane of parallel strips must do — with the whole of the driven track’s current returning through them.

Two checks run before anything is read. At direct current the shared resistance must be the plane’s whole resistance, the sheet resistance over its width, because the return spreads uniformly and both loops see all of it; the figures check that to a part in a million. And the transfer impedance must be the same whichever track is driven, which is reciprocity: the figures solve it both ways round at three frequencies and check agreement to a part in a billion rather than assume it.

All of the plane, then less of it #

With the tracks a millimetre apart the two loops share every one of the plane’s ten milliohms a metre at direct current. That is not a coincidence of the geometry. At direct current the return does not know where its track is: it spreads across the whole plane in proportion to the copper, and a voltage developed along the plane by that spread is a voltage along every loop that uses the plane. A sensor track a millimetre from a load track and one five centimetres away pick up the same shared resistance, and the plane has done nothing for either of them that a wire of the same resistance would not have done.

As the frequency rises the driven track’s return gathers under it, and the fraction of its own loop resistance that the quiet track shares falls. It passes a half at 183 kilohertz and settles, by ten megahertz, at 14.5 per cent. The inductance follows a similar course from a different start: at direct current the two loops share 39.8 per cent of the driven loop’s own inductance, because both loops enclose the same widely spread return, and above the band 3.41 per cent.

Those are fractions of the driven loop’s own impedance, and the driven loop’s own impedance changes across the band too — its resistance rises as its return crowds into a narrow strip of copper, which is the whole of the single-track essay’s result. So a shared fraction that falls does not mean a shared resistance that falls. At a millimetre the shared resistance itself rises from 10 milliohms a metre at direct current to 84 at 100 kilohertz before settling at 56. The plane shares less of a larger number.

The farther apart, the sooner it stops #

At three heights apart the sharing falls through a half at 525 kilohertz and settles at 32.7 per cent: the two returns, each gathered under its own track, still overlap by a third. At five heights it was 183 kilohertz and 14.5 per cent. At ten heights, two millimetres, the half is at 65.3 kilohertz and the resistance shared above the band is 4.03 per cent; at twenty-five, 20.9 kilohertz and 0.68; at fifty, 9.88 kilohertz and 0.18.

The half-frequency moves by a factor of fifty-three while the spacing moves by a factor of seventeen, and in the direction the single-track result predicts. A return does not gather under its track all at once; its outer parts leave first. The resistance two loops share is carried by copper that both returns still use, and for tracks far apart that is copper far from both, which is exactly the copper the returns abandon first. Tracks close together share the copper that stays in use longest, near both of them, and so go on sharing until much later in the band.

The practical reading is that separation buys independence twice. It lowers the resistance two returns share above the band, and it moves the frequency at which the sharing ends downwards, so a sensor routed far from a noisy track is isolated from the noisy track’s harmonics sooner as well as more. What it does not buy is anything at direct current, where the whole plane is shared by every track on it.

The single-track measurements these extend put the numbers in context. Where the current comes back estimated the single return’s change as one corner at 106 kilohertz, and the corner that is three decades wide solved it into a band from 1.59 kilohertz to beyond a megahertz; the half-sharing frequencies here, from 9.88 to 525 kilohertz, sit inside that band, which is where they must be. What the mutual inductance does above the band once it meets the tracks’ mutual capacitance — the two subtracting at the far end of the quiet track, and cancelling where their ratios match — is the far end that cancels, and how far the two may differ before the cancellation is gone, how wide a null is.

The inductance that changes sign #

The ten-millimetre figure carries something the one-millimetre figure does not: a mutual inductance that is negative. At direct current the two loops share −3.2 per cent of the driven loop’s inductance, and in the middle of the band that becomes −4.35 per cent at 17.8 kilohertz before rising through zero to +0.04 per cent above the band. At two millimetres the same thing happens more gently: +26.4 per cent at direct current, a dip to −2.10 per cent at 147 kilohertz, and +0.89 per cent above the band.

A negative mutual inductance means that a rising current in the driven loop induces a voltage along the quiet loop in the opposite sense to the one a naive picture of two parallel loops suggests. The naive picture has two loops side by side, each made of a track and the copper beneath it, with the flux of one threading the other in the same direction. That picture is right far above the band, where each return is under its own track and the loops are genuinely side by side.

In the middle of the band the driven loop’s return is neither spread nor gathered. Part of it has moved under its track and part of it is still out across the plane, and some of that spread part flows between the quiet track and the plane, or beyond the quiet track altogether. The quiet track’s loop then encloses some of the driven return’s flux in the opposite direction from the flux of the driven track itself, and when that return current is large enough the net linkage reverses. At ten millimetres, where the quiet track sits in copper the driven return is abandoning in exactly this part of the band, the reversal is the larger part of the coupling for most of a decade.

The cross-section cannot say whether that sign is visible in a real measurement, because it describes the inductance per metre of two long parallel runs and a real layout has ends, vias and components. What it does say is that a mutual inductance extracted at one frequency in the band and used at another can have the wrong sign, not merely the wrong size.

The quietest frequency #

The shared resistance and the mutual inductance add, as the real and imaginary parts of one impedance, and what a sensor sees is the magnitude.

For tracks a millimetre or two apart the coupling never falls below its direct-current value. The shared resistance does fall, as a fraction, but the mutual inductance’s reactance rises with frequency and overtakes it before the shared resistance has gone anywhere: at a millimetre the two are comparable by the middle of the band, and above it the coupling is inductive and rises a decade per decade, to 94.9 ohms a metre at a gigahertz.

For tracks five and ten millimetres apart there is a frequency at which the coupling is least, and it is lower than at direct current by a large factor. At five millimetres the voltage along the quiet loop falls from ten milliohms a metre to 2.66 near 501 kilohertz; at ten millimetres to 0.702, fourteen times less than at direct current. Below that frequency the coupling is the resistance of a return that has not finished gathering; above it, the mutual inductance of returns that have.

That minimum is the frequency at which a plane does the job it is usually credited with. A pair of tracks well apart is better isolated at half a megahertz than at direct current or at a gigahertz, which inverts the usual intuition that coupling simply rises with frequency. It also means that a test of isolation made at a convenient low frequency understates the coupling at direct current and overstates it in the middle of the band, and says little about either end.

A worked budget #

The figures are per ampere and per metre, which is the right normalisation for comparing geometries and the wrong one for deciding whether a particular layout is quiet enough, so it is worth running one number through.

Take a switching regulator’s input current, an ampere of ripple at five hundred kilohertz, on a track that runs twenty millimetres beside a sensor track carrying a signal of a few millivolts. Five millimetres apart, at the quietest frequency drawn, the coupling is 2.66 milliohms a metre, so twenty millimetres of run puts 53 microvolts along the sensor’s loop for each ampere. At a millimetre apart the coupling at the same frequency is well above its direct-current ten milliohms a metre, so the same run puts more than two hundred microvolts there. Moving the sensor track four millimetres further away is worth a factor of nearly four at the very least at the switching frequency, before the harmonics are counted.

The harmonics are where the plane’s help runs out. The regulator’s edges carry energy far above its switching frequency, and above the band the coupling is inductive and rises a decade per decade: at a gigahertz, 4.05 ohms a metre at five millimetres and 94.9 at one, which for twenty millimetres of run is 81 and 1,900 milliohms for each ampere of that harmonic. A fast edge on the load track is therefore coupled into the sensor through the mutual inductance of two image currents, in exactly the proportion the closed form below gives, and the separation that halved the switching-frequency coupling reduces the edge coupling by the ratio of the two image formulas — a factor of twenty-three between one and five millimetres.

None of that is small enough to ignore on a sensor reading a few millivolts, and none of it is the direct-current coupling a plane is usually credited with removing. What the plane removes is the shared wire: the ten milliohms a metre it leaves at direct current is a small fraction of what a dedicated return conductor of ordinary size would have shared, and the separation-dependent coupling above it is a small fraction of what two unshielded loops would have. It removes neither completely, and it trades them against each other across the band.

Above the band, in closed form #

Far above the band each return is the image of its track, a distribution across the plane whose density falls as $h/(\pi(h^2 + x^2))$ from the point under the track. Two such distributions of half-width $h$, a distance $d$ apart, overlap in a distribution of half-width $2h$, and the shared resistance should be that overlap.

It is. The figure checks the shared resistance against $4h^2/(4h^2 + d^2)$ at every spacing drawn, to fifteen per cent, and it lands closer than that at every spacing but the widest: 32.7 per cent against 30.8 at three heights, 14.5 against 13.8 at five, 0.68 against 0.64 at twenty-five. At fifty heights the finite plane holds the tails in and the solve reads 0.18 against 0.16.

The mutual inductance has its own closed form for filaments over a perfect plane, $(\mu_0/4\pi)\ln(1 + (2h/d)^2)$, and the solved value is within five per cent of it at every spacing: 15.1 nanohenries a metre against 14.8 at a millimetre, 0.645 against 0.638 at five. The residual is the tracks’ width, which the filament formula ignores and the strips do not.

So above the band the plane does what the textbook says, in the textbook’s own formula. The shared resistance falls as the square of the spacing once the spacing is a few heights, and the mutual inductance as the square as well. What the textbook formula does not contain is the rest of the band — the whole plane shared at direct current, the sign change in the middle, and the quietest frequency — and a coupling budget built on the formula is right above ten megahertz and nowhere below it.

What the cross-section says, and what it does not #

The plane is shared, completely, at direct current. Every track on a plane shares its whole resistance with every other at direct current, and a plane of fifty millimetres of half-milliohm copper is ten milliohms a metre of it. That is smaller than a wire, not absent. The millivolts in the wire found half a millivolt from a hundred milliamps through five milliohms of shared wire; the same hundred milliamps along ten millimetres of run over this plane puts ten microvolts along every other loop on it at direct current, whatever the routing.

Separation works in the band and above it, and works twice. It lowers what is shared above the band as the square of the spacing and moves the end of the sharing down in frequency.

It is a cross-section. The tracks are infinitely long and parallel, the plane is uniform along them, and nothing leaves the board. Real returns leave their tracks at vias and components, and where a return has to travel along the plane between two points the geometry varies along the run — which is the crossing-slot problem that where the plane runs out could not pose in a cross-section either. The numbers here are per metre of parallel run, and they are the right numbers for two long tracks and a lower bound for anything with ends.

The tracks are identical and at the same height. A sensor track on a different layer from the load’s, or much narrower, has a different image distribution and a different overlap, and the closed form changes with it.

Still open: a plane with an edge between them, a split between them, and the ends of the run #

Two tracks near an edge. Where the plane runs out found a return crowding into the last few hundred micrometres of copper when its track runs near the plane’s edge. Two tracks near the same edge crowd into the same copper, so their shared resistance above the band should be larger than the image overlap, and their sharing should end later in the band. The same cross-section with an edge in it would put numbers on how much of the isolation an edge takes back.

A slot between them. Cutting the plane between two tracks is the common response to coupling, and in a cross-section it separates the two returns completely at direct current — each track’s return can use only its own side. Whether the slot helps in the band, where each return has already left the far copper, or only at direct current, and what it costs each track’s own loop inductance, is the measurement that would decide whether splitting a plane is worth its risks.

The ends of the run. Per metre of parallel run is the right unit for long buses and the wrong one for a sensor track that runs a centimetre beside a switching node and then leaves. The coupling of a short run is dominated by where each return enters and leaves the plane, which a cross-section cannot describe and a two-dimensional solve of the plane’s surface currents could.