The slot that does not separate

Two returns in one plane put two tracks over one plane and measured what their returns share. At direct current they share all of it: each return spreads across the whole plane’s width, so the plane’s 10 mΩ per metre is common to both loops however far apart the tracks are routed. The sharing ends across a band a few decades wide, as each return gathers under its own track, and above the band two tracks a millimetre apart share a few per cent of each other’s resistance and a little of their inductance.

That essay ended on the repair almost every board designer reaches for first. Cutting the plane between two tracks looks, in a cross-section, as if it separates their returns completely at direct current — each return can use only its own side — and the question is 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. The measurement says it does not help at either, when the plane is one ground, and it says what does.

A slot in one plane #

The cross-section is the earlier essay’s: two tracks 200 µm wide and 200 µm above a 50 mm plane of 0.5 mΩ a square, 2 mm apart. One is driven and the voltage along the other’s loop is read, with the plane cut into strips whose currents are solved together. The new element is a slot along the plane, centred between the tracks and running the length of the parallel run.

At direct current the slot changes nothing: the share of the plane’s resistance the two loops have in common is 1.0000 with the slot as without it. The reason is what a slot in one ground plane is. The two halves are one conductor, joined wherever the slot ends — at the ends of the parallel run, at the edges of the board, through every via that ties the plane together — and a current spreading along a run of any length uses both halves, in parallel, exactly as it used the uncut plane. The cross-section shows a gap; the circuit sees two strips of copper connected at both ends.

Above the band it is worse than no slot: the share rises from 4.03 per cent to 4.98. Each return has gathered under its own track by then, and its outer edge — the part of the return nearest the other track — lies where the slot now is. With that copper removed the return crowds against the slot’s edge instead, which is nearer the other track’s return, and the two share more. The slider on the figure at the head of the page widens the slot from a fifth of a millimetre to 1.6, and the in-band share rises with it.

The width of the slot #

At 100 MHz the one-plane share rises with every increase in the slot’s width: 4.03 per cent uncut, 4.37 at a fifth of a millimetre, 4.95 at 0.4, 6.88 at 0.8, and 15.7 per cent at 1.6 mm, where the slot has taken most of the copper between tracks 2 mm apart. The pattern is the one where the plane runs out found for a single track near an edge: a return that cannot spread where it wants to crowds into the last copper it has, and an edge — whether of a board or of a slot — is where that crowding happens. A slot between two tracks is two edges, each beside the other track.

So in a single ground plane a slot between two tracks makes their coupling through the plane’s resistance worse in the band, by a quarter at a modest width and by nearly four times at a wide one, and leaves it untouched at direct current. It is the wrong repair for the problem it is usually applied to.

Two grounds #

The slot does separate the returns if the halves are genuinely separate: two ground nets, each joined only to itself, meeting at no via and no edge along the run. The cross-section is the same; the circuit is different, because now each track’s current must return in its own half.

Split, the loops share nothing at direct current — the solve returns a share of $3 \times 10^{-21}$ — since no current from one track can flow in the other’s half. This is what the slot was supposed to do, and it does it only when the halves are different nets.

It does not stay at nothing. As the frequency rises the driven loop’s magnetic field reaches the quiet half and induces a circulating current there, which the quiet loop sees as a voltage in phase with the driving current: the share rises to a peak of 27.2 per cent near 26 kHz, in the middle of the band where the returns are gathering, and settles at 1.54 per cent above it. The split plane shares less than the uncut plane at high frequency, 1.54 against 4.03 per cent, but it shares something, and in the middle of the band it shares a great deal more than either of the others.

What the split costs #

Separating the returns is not free, and the cost is paid by each track’s own loop.

At direct current each loop’s own return resistance is the whole plane’s 10 mΩ per metre over an uncut plane, 10.1 with the slot, and 21 mΩ per metre split — 2.10 times, the ratio of the plane’s width to the half the track sits over. A signal that shared its return with its neighbour now has half the copper to itself. Above the band the three agree, since each return has gathered under its own track and never reached the slot, and each loop’s own inductance is 443 or 444 nH per metre in all three, the slot and the split costing a third of a per cent at most.

So a split trades shared resistance for own resistance at direct current, which is a good trade when the shared resistance is the problem — a sensitive measurement returning through the same copper as a load’s current — and a bad one when each loop’s own drop matters. A star is half a millimetre long found the same trade in the other direction: a star point removes shared impedance by giving each return its own path, and the cost is the length of those paths.

Why the split peaks in the middle of the band #

The peak is the least intuitive number here, and it has a plain cause. Split, the quiet half cannot carry any of the driven track’s current as a net current, since its net is not the driven track’s return. What it can carry is a current that flows one way along the half near the slot and back along the far side of the same half — a circulating current, summing to zero, driven by the changing magnetic field of the driven loop next door. At direct current there is no changing field and no such current. At high frequency the half’s own inductance limits it, and the circulation settles into a thin sheet that shields the quiet loop a little and couples it a little. In between, where the reactance of that circulating path is comparable with the resistance of the half it flows in, the induced current is largest in proportion to its cause, and so is the voltage it leaves along the quiet loop.

The arithmetic puts that crossing where the solve found it. The half has 21 mΩ per metre of resistance along its length; a reactance equal to that at 26 kHz is an inductance of about 130 nH per metre — less than a third of each loop’s own 443 nH per metre, which is what a circulating path confined to one half of the plane, with no track above it, ought to have. The peak is not a resonance and it is not sharp: it is the familiar shape of an eddy current in a conductor whose resistance and inductance meet at one frequency, and the solve places it inside the same band in which the corner that is three decades wide found a single return gathering under its track. Both are one question — where the plane’s resistance stops deciding the currents and its inductance starts — asked of two different current paths.

What makes the peak matter is that the band it sits in is the band a switching converter’s ripple and harmonics occupy. A split plane that separates a converter’s return from a measurement’s return at direct current, and is then asked to keep them apart at 30 kHz, is being asked at the one frequency where it is worst.

An ampere, ten centimetres, and a millivolt #

The figures are per metre and per ampere, which hides how small or large they are. Take a run ten centimetres long with an ampere of steady load current in the driven track and a precision sensor’s return under the quiet one. Over an uncut plane the sensor’s loop sees 10.2 mΩ per metre of shared resistance, so 1.02 mV is added to its reading — about the size of the error the millivolts in the wire found for a hundred milliamps returning through a centimetre of track. Slotted, the same 1.02 mV, since the slot changed nothing at direct current. Split, the steady ampere adds nothing at all; the 1.26 mΩ per metre left at a kilohertz is the tracks’ mutual inductance, about 200 nH per metre, and a steady current in an inductance induces no voltage.

At 100 MHz the numbers are of a different kind. The driven track’s current there is an edge’s harmonic rather than a load, and ten milliamps of it over ten centimetres puts 2.5 mV on the quiet loop over an uncut plane and 1.75 mV split. The split has bought three quarters of a millivolt on a signal the layout was not designed to separate, and it has doubled the steady drop in both loops to buy the millivolt at direct current. For a designer, the useful form of the result is that ratio: a split is a direct-current repair that is paid for in direct-current resistance, and at high frequency it is neither a repair nor much of a cost.

The coupling a layout is judged by #

What a designer finally cares about is the voltage one track’s current puts along the other’s loop — the transfer impedance, which combines the shared resistance and the mutual inductance.

Uncut or slotted, the coupling at low frequency is the shared plane resistance, 10.2 mΩ per metre at a kilohertz, and at high frequency the tracks’ mutual inductance, 2.48 Ω per metre at 100 MHz. Split, the shared resistance is gone, and what remains at a kilohertz is the tracks’ own mutual inductance, 1.26 mΩ per metre, eight times less. At 100 MHz the split’s coupling is 1.75 Ω per metre, only 29 per cent less than the uncut plane’s.

That is the reading that settles the question. In the band the two tracks couple mainly through their own magnetic fields, which extend over each other whatever copper is or is not beneath them; the plane shapes those fields but does not carry them, and cutting it changes them by a fraction. The far end that cancels measured coupled tracks’ crosstalk as a property of their mutual inductance and capacitance; a slot in the plane under them does not change what that essay measured, except to spread each field a little further and couple the tracks a little more.

What a designer should take #

Do not cut a slot between two tracks to separate their returns in a single ground plane. At direct current it separates nothing, because the halves are joined at the ends of the run; in the band it makes the shared resistance worse, by a quarter at 0.4 mm and nearly fourfold at 1.6 mm, by removing the copper each return would have spread into.

If the returns must be separated at low frequency — a precision measurement sharing a board with a power load — give them separate grounds, joined at one point away from the run, and accept each loop’s doubled resistance. Expect the separation to hold at direct current and to fail in the middle of the band, where the split shares more than an uncut plane would, and to buy little at high frequency, where the tracks’ own fields do the coupling. For high-frequency isolation, distance between the tracks, and the image-current law where the current comes back began with, are what work. A differential pair routed over the slot is a different case again, since its two returns are meant to cancel and a slot changes how equally they do — the question the millimetre that becomes common mode asked of a length mismatch.

How the numbers were obtained #

The cross-section is a set of copper strips in the plane under two filament-width tracks, each strip’s partial self and mutual inductances computed from the geometric mean distances of flat strips, and each strip’s resistance from the sheet resistance. At each frequency the strip currents are solved with the driven track’s current as the constraint on the total return; for one ground plane every strip shares one longitudinal voltage, and for a split plane each half has its own and must carry the driven current or none, according to which track is over it. Strips are graded finely under each track and at each panel’s edges. With one uncut panel the solve agrees with the earlier two-track solve, built on a different discretisation, to a per cent across the band. The voltage on the quiet loop comes from the strip currents and the partial inductances between them and that loop.

What it leaves out #

The ends of the run. A cross-section describes a long parallel run, and it assumes a slotted plane’s halves are joined where the run ends. How far from the ends that holds — how long a slot must be before its middle behaves as this cross-section — is a question about current spreading along the run that a cross-section cannot answer.

A slot that crosses a track. Every slot here runs along the tracks. A slot crossing under a track forces the return to detour round the slot’s end, which adds a loop and an inductance that has nothing to do with the effects here and is usually much larger.

The capacitive coupling. Everything here is the plane’s resistance and the tracks’ inductance. The electric coupling between the tracks depends on the copper beneath them too, and a slot removes some of the capacitance each track has to ground, which raises their capacitance to each other relatively; that half of the crosstalk is not solved.

Still open: the length of a slot, a guard trace instead, and the split joined at one point #

How long a slot has to be. The one-plane result assumes the halves are joined far from where the coupling is measured. A slot of finite length, joined at both ends a stated distance from the measurement, would show the direct-current sharing return as the ends approach — a two-dimensional solve of the plane’s surface currents rather than a cross-section.

A grounded guard trace between the tracks. Instead of removing copper between the tracks, a designer can add it: a grounded trace between them. Whether that reduces the in-band shared resistance and the mutual inductance, where the slot increased the first and left the second, is the same cross-section with a third conductor in it.

Two grounds joined at one point. A split plane is usually joined somewhere, at a star point. With the join a stated distance from the run, the direct-current separation holds only as well as the join’s own impedance allows, and one voltage added to two readings has the arithmetic for what that join then contributes to both.