The guard that is part of the return

The slot that does not separate cut a slot in the plane between two tracks, the usual response to coupling between their returns, and found it does the opposite of what it looks like it does. In one ground plane the halves stay joined where the run ends, so at direct current both returns still spread across the whole plane and share all of its resistance; and above the band the slot makes the sharing worse — 4.98 per cent of one loop’s resistance carried in the other’s return against 4.03 with no slot, rising to 15.7 as the slot widens to 1.6 mm — because it removes the copper each return would have spread into.

It ended with the obvious inversion: 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.

The cross-section with a guard in it #

The geometry is the earlier essays’: two tracks 200 µm wide, 200 µm above a 50 mm plane of half a milliohm a square, their centres 2 mm apart. One track is driven, the other listened to, and the quantity is the same: the share of the driven loop’s resistance that the quiet loop’s return also carries — the resistance they have in common, over the driven loop’s own. It is the coupling a common impedance makes between two circuits that share a return.

The guard is a trace of the same copper at the tracks’ own height, centred between them, and joined to the plane at both ends of the run — by vias at the ends, which is the ordinary way a guard trace is grounded. That makes it part of the return net: it carries the same longitudinal voltage as the plane and a share of the return current. The cross-section is solved as before, with every strip of plane and every strip of guard a filament carrying its own current, coupled to every other by partial inductance and each with its own resistance, and the whole return net held at one voltage per unit length. The guard is cut into eight strips of its own so that its current can crowd to the side nearer the driven track.

With no guard the new solve is the earlier two-track solve to a part in $10^9$ at every frequency drawn, which is the check that the guard has been added to the old problem rather than a new problem written.

Nothing at direct current #

Every curve starts at exactly one. A guard joined to the plane at both ends is, at direct current, a strip of copper in parallel with the plane: both returns spread across the plane and the guard together in proportion to their conductance, and the resistance they share is the whole of the parallel combination. Adding a trace to the return net cannot separate two currents that both flow in the return net, any more than the slot could; only separate nets can, and the slot that does not separate showed what separate nets cost each loop.

So the guard is a high-frequency measure or nothing. That is not a defect of this guard or of this geometry: it follows from the guard being joined to the plane, and every grounded guard is joined to the plane somewhere, because a guard that is not grounded is a floating conductor that couples the two tracks capacitively instead. Below the band where each return gathers under its own track — the band that two returns in one plane found ending near half a megahertz for tracks this far apart — the guard changes the sharing by less than a per cent.

Above the band, a little #

Above the band each return has gathered under its own track, and what the two loops still share is the overlap of those distributions — 4.03 per cent of the resistance with nothing between the tracks. A guard 0.6 mm wide moves it to 3.50 per cent; 1.2 mm, to 2.84. The mutual inductance falls further: from 0.89 per cent of the driven loop’s own inductance to 0.68 with the 0.6 mm guard and 0.52 with the 1.2 mm one.

Against the guard’s width, at 100 MHz, the shared resistance goes 4.03, 4.44, 4.09, 3.74, 3.50, 3.29, 3.07, 2.84 and 2.55 per cent for guards from nothing to 1.4 mm — the last leaving 0.2 mm on each side between it and the tracks — and the mutual inductance falls from 0.0089 of the self to 0.0044. The slot, drawn beside it, raises the sharing from 4.03 to 10.45 per cent over the same widths.

So the inversion holds in direction: copper added between the returns helps and copper removed hurts. It does not hold in size. A guard filling three-quarters of the gap removes a little over a third of the in-band shared resistance; a slot 1.2 mm wide more than doubles it, to 10.45 per cent. The two are not symmetric, because the slot removes copper the returns were using while the guard adds copper they were not.

Why a thin guard is worse than none #

The first two points of that curve go the wrong way. A guard 0.1 mm wide raises the shared resistance from 4.03 to 4.44 per cent, and one 0.2 mm wide to 4.09; only from about 0.3 mm does the guard begin to help.

The reason is that the guard is shared by construction. It sits between the two tracks at their height, closer to each than the plane beneath the other track is, so each return sends some of its current along it. A thin guard has a high resistance per unit length — half a milliohm a square across a fifth of a millimetre is two and a half ohms a metre, against the plane’s ten milliohms — and a current that both returns push through a high resistance is exactly a shared impedance. As the guard widens, its resistance falls, and its other effect — pulling each return away from the plane beneath the other track — takes over. The crossover, near a third of a millimetre here, is where the two effects balance.

That is worth knowing because the thin guard is the common one: a trace of the minimum width the board allows, run between two signals to “isolate” them, is on the wrong side of this curve.

Where the guard’s benefit comes from, and its limit #

The guard carries a share of the driven track’s return: 1.2 per cent at direct current — its width against the plane’s, 0.6 mm in 50 — and 7.0 per cent above the band. That is how it helps. Every ampere of the driven return that flows in the guard is current that would otherwise have spread through the plane towards and under the quiet track, which is where the shared resistance lives. A 1.2 mm guard takes 15.7 per cent above the band; a 0.2 mm one, 3.5.

It is also why the guard’s benefit is bounded. The shared resistance is the overlap of the two returns’ distributions in the plane, and most of each return is directly beneath its own track, far from the guard and far from the other return. The guard can intercept only the part of the distribution that has spread sideways towards it — the tail that where the current comes back found falling as $1/(1 + (x/h)^2)$ across the plane — and that tail is a minority of the current. A guard that took all of the tail would still leave the overlap under each track, and that is what the curve is approaching as the guard widens: not zero.

The slot, for comparison #

The slot and the guard are the two moves a layout offers between two tracks, and the cross-section treats them as what they are: one removes copper from the return net and the other adds it. Neither touches the direct-current sharing, because both leave a single connected return net. Above the band the slot pushes each return towards the slot’s edge beside the other track, and the guard pulls part of each return onto a conductor between them and away from the plane beneath the other track. The first increases the overlap of the two distributions and the second reduces it. The only arrangement that removes the direct-current sharing is two separate nets, and the earlier essay’s split plane paid for that with each loop’s own resistance and inductance.

Distance, which does more #

The alternative to a guard is room. Two returns in one plane found the in-band overlap falling as $4h^2/(4h^2 + d^2)$ with the tracks’ separation $d$ — 14.5 per cent at three heights apart. At ten heights, the geometry here, it is about 4 per cent, the unguarded figure. Doubling the separation to twenty heights would bring it to about one per cent, which is what no guard in a ten-height gap achieves.

So the guard is what to reach for when the tracks cannot move and a modest reduction is enough; distance is what to reach for when a large one is needed. The two combine: a guard in a wider gap has more room to be wide without crowding either track, and the curve above says its benefit grows with its width.

What a layout takes from this #

A guard trace is worth having only if it is wide. Below about a third of a millimetre in this geometry it adds a shared resistive path and makes the coupling slightly worse; above that it helps in proportion to how much of the gap it fills. A minimum-width guard between two signals is a habit this cross-section argues against.

It does nothing for low frequencies. Below the band in which each return gathers under its own track — the corner that is three decades wide is how wide that band is — the two returns spread across the whole plane whatever lies between the tracks. A guard is no remedy for the shared resistance that couples a large low-frequency current into a small low-frequency measurement; only separate returns are.

Its benefit is a fraction, not an order of magnitude. A third of the in-band shared resistance and half the mutual inductance is what a guard filling most of a ten-height gap buys. A factor of four needs twice the separation. Where a design needs an order of magnitude, the guard is at best a supplement to room.

And the plane’s own edge matters more. Where the plane runs out found that a plane ending under or near a track costs a quarter of the loop’s inductance and changes where its return can go; a guard near the edge of a plane does nothing about that, and a layout that has both problems should fix the edge first.

The coupling a guard is better at #

The cross-section here measures the coupling through the return: shared resistance and mutual inductance, the magnetic and resistive paths. Between two tracks there is also an electric path, their mutual capacitance, and a grounded conductor between them intercepts field lines that would otherwise have reached from one track to the other. That is the coupling a guard is usually drawn for, and it is the one this solve does not compute.

For signals at high impedance the capacitive path dominates, and a grounded guard is a strong remedy for it — the reason guard traces are common in high-impedance analogue layouts. For signals at low impedance, where the return current is large and the coupling is through the shared return, the guard is the modest remedy measured above. The far end that cancels is where the capacitive and inductive couplings between two lines are set against each other, and a guard changes both of them by different amounts; which one a design is suffering from decides whether a guard is worth its width.

How the cross-section was solved #

The plane and the guard are strips, each a filament with its own resistance from the copper’s sheet resistance and its own current, coupled to every other strip and to both tracks by partial inductances computed in closed form from the mean logarithmic distance between two intervals — coplanar for two strips at one height, offset by the height for a guard strip against a plane strip. The return net’s strips share one longitudinal voltage, and the driven track’s current returns entirely in that net.

The quantities are the loop impedances between the tracks: the shared resistance is the real part of the transfer impedance, the mutual inductance its imaginary part over the frequency, each against the driven loop’s own. Checked: with no guard the solve equals the earlier two-track solve to a part in $10^9$; with any guard, the whole plane’s resistance is shared at direct current to a part in a million; the guard’s direct-current share of the return equals its width over the plane’s to two per cent; and a guard filling most of the gap lowers both the shared resistance and the mutual inductance, while a slot of any width raises the shared resistance.

A guard joined only at its ends, in two dimensions #

The guard is joined only at the ends. A guard stitched to the plane by vias along its length is held at the plane’s potential everywhere, which turns it from a parallel return into something closer to a wall; at high frequency, where the vias’ own inductance matters, the spacing of the stitching becomes the variable.

The cross-section is two-dimensional. The run is long against the separation, and the ends where the guard and the halves of any slot join are not in the cross-section; a short run, or a guard shorter than the tracks, is a three-dimensional problem.

The tracks are single-ended. A differential pair carries its own return in its partner, and the plane’s share of its return is the common-mode part only; the millimetre that becomes common mode is how a small asymmetry creates that part. A guard between two pairs acts on their common-mode returns exactly as it acts here, and on their differential signals hardly at all, so for pairs the guard’s value is tied to how much common mode they carry.

And the coupling is resistive and inductive only. A guard between two tracks also changes their mutual capacitance, which is the coupling that dominates at high impedance, and which a grounded guard reduces far more effectively than it reduces a shared return.

Still open: the stitched guard, the join at one point, and the length of a slot #

A guard stitched along its length. Vias every few millimetres hold the guard at the plane’s potential along the run. Whether that lowers the shared resistance further — or raises it, by joining the guard’s high-resistance path more firmly to both returns — depends on the spacing, and a cross-section cannot answer it: it needs the run in three dimensions.

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 contributes to both.

How long a slot has to be. The earlier 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.