Lines, where a wire has a length

Where the plane runs out

The corner that is three decades wide solved a return current over a plane that extends well past the track on both sides. Where it does not, the two costs arrive at opposite ends of the band: at direct current a plane that ends under the track costs 27 per cent of inductance and not one part in a million of resistance, and above the band it costs 16 per cent of inductance and 199 per cent of resistance. Three track-heights of copper beside the track removes almost all of both, and the number three has no millimetres in it — sixteen times the whole cross-section gives the same ratios to a part in a billion.

Assumes: Where the current comes back · Kirchhoff's own frequency

The corner that is three decades wide solved a return current across a cross-section of its plane and found a band where where the current comes back had estimated a corner. Every number in that solve came from a plane fifty millimetres wide with the track in the middle of it, twenty-five millimetres of copper on either side, and the return free to use as much of it as the impedance matrix wanted.

A board is not usually like that. A track runs near the edge of its plane, or past it into a region the plane does not cover, or over a slot cut along it for a connector or a crossing. The question this essay answers is what that costs, and it is a question about a boundary rather than about a frequency: not when does the return gather but where is it allowed to be.

The answer separates into two numbers that behave in opposite ways, and neither of them is what moving copper sideways suggests. At direct current a plane that ends directly under the track costs twenty-seven per cent of the loop’s inductance and exactly nothing in resistance. Far above the band it costs sixteen per cent of inductance and very nearly triples the resistance. The two costs arrive at opposite ends of the same band the corner that is three decades wide measured, and the one every designer worries about — the inductance — is at its worst where nobody looks for it.

A track needs about three heights of copper beside it, and it is the resistance that says so. computed by solving, not by drawing at 100 MHz, each point a strip solve of its own on a 50 mm plane of the same area, moved sideways. The horizontal axis is where the track's centre sits relative to the plane's edge, in units of the track's height above it; negative is a track hanging past the edge with no copper beneath it. With the centre directly over the edge the loop's inductance is 1.161 times its centred value and its resistance 2.90 times, because the return has to crowd into the last few hundred micrometres of copper. Three heights in, the inductance is 1.006 times and the resistance 1.09; ten heights in, both are within 0.7 per cent. Three heights past the edge the inductance is 1.82 times. At direct current every point on this axis is exactly one, because the copper has been moved and not removed. — Fig. 1 The loop’s inductance and resistance at 100 MHz against where the track’s centre sits relative to the plane’s edge, in units of the track’s own height above it, each divided by the same quantity for a centred plane of the same area. Negative is a track hanging past the edge with nothing beneath it. Over the edge the inductance is 1.161 times and the resistance 2.90; three heights in, 1.006 and 1.09; ten heights in, both within 0.7 per cent.

The copper is moved, and at first nothing happens

The geometry is the one that three-decade band was measured on, and nothing else has changed. The plane is still fifty millimetres of half-milliohm copper, the track is still two hundred micrometres square and two hundred micrometres above it, and the strips are still joined at both ends so that the return divides itself between them however the impedance matrix says. What is different is where the copper is: the whole plane is slid sideways, so that the track’s centre sits a stated number of its own heights inside the plane’s edge.

That is a move and not a removal, and the direct-current resistance says so exactly. It is the sheet resistance over the width of copper, and the width of copper has not changed, so the loop still meets 1.000 milliohms over a hundred millimetres wherever the plane is put — to twelve decimal places, which is the solve returning an identity rather than being told one.

The inductance at direct current does change, and it is the first surprise. At direct current every strip of the plane carries the same current per unit width, because the inductances have dropped out of the impedance and only the resistances are left. A uniform return spread over copper lying entirely to one side of the track is a bigger loop than the same uniform return spread symmetrically around it: its centre of current is twenty-five millimetres away instead of underneath. Solved, the loop is 130.68 nanohenries over a hundred millimetres against 103.20, which is 1.266 times, and that is the largest inductance penalty an edge ever imposes.

So the picture a designer carries — that a plane’s edge is a high-frequency problem — has the sign of the inductance term backwards. The inductance penalty is worst at direct current and gets better as the frequency rises, because the gathering measured on a centred plane pulls the return in under the track and, once it is there, it no longer matters much where the far edges of the copper are.

An edge's two costs arrive at opposite ends of the band. computed by solving, not by drawing. The loop's inductance and resistance for a track whose centre is 0, 1 and 3 heights inside the edge of its plane, each divided by the same quantity for a centred plane of the same area, against frequency. At direct current the resistance ratio is 1.000000 — exactly one, because the copper has been moved sideways and not taken away — and the inductance ratio is already 1.266, its largest value anywhere, because a return spread evenly over copper that lies to one side encloses more flux than the same copper spread around the track. As the return gathers the two exchange places: above the band the resistance is 2.99 times and the inductance 1.160. The curves for one and three heights in lie between and below. — Fig. 2 The same two ratios against frequency, for a track 0, 1 and 3 heights inside the edge. At direct current the resistance ratio is 1.000000 and the inductance ratio is already 1.266; above the band the resistance is 2.99 times and the inductance 1.160. The two costs cross over inside the band the return gathers across.

The resistance arrives with the gathering

The resistance penalty is the other way round, and its mechanism is visible rather than inferred.

Below the band the return is spread over all the copper there is, so it meets the same resistance whether that copper is centred or not. As the frequency rises the return tries to gather beneath the track, and on a centred plane it can: the image-current distribution has half its current within one track-height of the point under the track and tails away symmetrically. With the plane ending under the track’s centre, half of that distribution has nowhere to go. The current it was going to carry does not disappear; it is pushed into the copper that remains, nearest the edge, where it crowds into the last few hundred micrometres.

Solved at a hundred megahertz, the peak current density in the copper beside the edge is 14.0 amperes per millimetre of plane for one ampere in the track, against 1.48 on a centred plane — nine and a half times. That crowding is the resistance: 111.26 milliohms over a hundred millimetres against 38.31, and above the band, where the gathering is complete, 2.99 times rather than 2.90.

The measurement worth keeping from this is not the factor of three. It is that the resistance penalty is a current-density result and the inductance penalty is a loop-area result, and the two live at opposite ends of the band because the return’s position is what sets both and it changes across the band. The centred solve found the gathering takes about three decades; this one finds that the two consequences of an edge are separated by exactly that span.

Fig. 3 Current per millimetre of plane against distance sideways from the point under the track, measured in track heights, for one ampere in the track, at 100 MHz. The shaded region has no copper. The density in the copper next to the edge reaches 14.0 A/mm against 1.48 on the uncut plane. The slider is the frequency.

The same picture at a kilohertz, where nothing is crowded

Drawn at a kilohertz the edge looks harmless, and looking harmless is the point.

At a kilohertz the return is still essentially uniform. The density in the strip beside the edge is within a few per cent of the density twenty millimetres away, there is no crowding to see, and the resistance ratio is 1.007 — seven parts in a thousand, which nothing would ever notice. A measurement taken there, or a solve stopped there, says an edge costs nothing.

The inductance at that frequency is 1.265 times its centred value, which is twenty-six per cent, and it is invisible in the same picture because inductance is not a local quantity. It is set by where the current is on average, and the average is twenty-five millimetres off to one side whether the distribution is crowded or flat. The two numbers a cross-section reports about the same geometry therefore disagree about whether anything is wrong, and both are right.

This is the shape that every model has an edge keeps finding and that the edge that is a region named: a boundary that is crossed in one quantity and not in another, so which side of it a design is on depends on what the design is sensitive to. A circuit whose loop inductance sets its ringing is already in trouble at a kilohertz. A circuit whose shared copper sets a voltage error — the situation the millivolts in the wire measures — is not in trouble until the band.

The return where the plane ends directly under the track. computed by solving, not by drawing at 1.00 kHz. Current per millimetre of plane against distance sideways from the point beneath the track, measured in track heights, for one ampere in the track. The heavy curve is the cut plane and the faint one the same track over a centred plane of the same area; the shaded region is where there is no copper. The density in the copper nearest the missing part rises to 2.39e-2 A/mm against 2.22e-2 on the uncut plane. The return crowds into the copper nearest the missing part, which is where the extra resistance comes from — the current is not spread over less copper by choice but by there being none. — Fig. 4 The same cross-section at a kilohertz. The two curves lie almost on top of each other: the return has not gathered, so there is nothing for the edge to interrupt, and the resistance ratio is 1.007. The loop’s inductance at this frequency is nevertheless 1.265 times its centred value, because inductance is set by where the current is on average rather than by how sharply it is peaked.

Three heights, and the number has no millimetres in it

The useful result is how quickly both penalties disappear once there is copper beside the track.

Measured above the band, where the resistance penalty is largest, the resistance ratio is 1.663 with the track one height inside the edge, 1.207 at two heights, 1.090 at three, 1.048 at four and 1.030 at five. The inductance ratio over the same range is 1.038, 1.013, 1.006, 1.004 and 1.002. Three track-heights of copper beside the track takes the inductance penalty below one per cent and the resistance penalty below ten, and ten heights takes both below one.

The interesting part is the unit. Three heights is not three of anything else, and the claim that it is the right unit is testable: scale the whole cross-section — the track’s height, the track’s width, the plane’s width and the inset, all by one factor — and read the ratios at the same point in each geometry’s own band. If what an edge costs had a length in it, the curves would separate.

They do not separate. Over a sixteenfold range of size, from a track fifty micrometres above its plane to one eight hundred micrometres above it, the two ratios agree to a part in six billion. That is not a fit and not an approximation: the strip equations contain no length except through the geometry’s own proportions, so the whole family of cross-sections is one cross-section drawn at different magnifications, and the solve recovers that without being told.

So the design rule is a ratio and can be stated once. A four-layer board with two hundred micrometres between signal and plane wants six hundred micrometres of copper beyond the track. A two-layer board with 1.6 millimetres between them wants five millimetres, which is a great deal more copper for the same rule and is the sort of thing a stack-up decides silently.

What an edge costs has no length in it. computed by solving, not by drawing. The same picture at three sizes: a track 50 µm, 200 µm, 800 µm above its plane, with the track's width, the plane's width and the inset all scaled in proportion, and the track's centre directly over the plane's edge in each. The horizontal axis is frequency divided by Rs/2πµ₀h for that geometry, which is the only frequency the cross-section contains. The three sets of curves lie on one another to 1.6e-10 across a sixteenfold range of size — so a designer's question is how many track-heights of copper lie beside the track, and never how many millimetres. — Fig. 5 The same geometry at three sizes — 50, 200 and 800 µm above the plane, with the track’s width, the plane’s width and the inset scaled in proportion — against frequency divided by that geometry’s own Rs/2πµ₀h. Three sets of curves, one pair of lines: they agree to 1.6 × 10⁻¹⁰ across a sixteenfold range of size.

Past the edge, where the loop stops having a bottom

A track that runs off the end of its plane altogether is a different object, and the sweep reaches it by continuing past zero.

Three heights past the edge — six hundred micrometres of board with no copper under the track at all — the loop’s inductance is 1.82 times its centred value, and ten heights past it is 2.36 times. Those are large numbers by the standards of everything else here, and they are also smaller than a designer expects, which is worth saying plainly: a track leaving its plane does not immediately acquire the inductance of a wire in free space, because the plane is still there, a little to one side, and the return still uses it.

The resistance past the edge does something the inductance does not: it falls again. At one height past the edge it is 2.93 times the centred value, at three heights 1.61, and at ten heights 0.61 — below the centred plane. The reason is that a return which cannot get under the track has no reason to crowd, so it spreads along whatever copper it can reach, and a spread return is a low-resistance one. The loop is worse in every way that matters and its resistance is better, which is the clearest statement this field has that resistance and inductance are not two readings of one quality.

The return under a track 3 heights past the plane's edge. computed by solving, not by drawing at 100 MHz. Current per millimetre of plane against distance sideways from the point beneath the track, measured in track heights, for one ampere in the track. The heavy curve is the cut plane and the faint one the same track over a centred plane of the same area; the shaded region is where there is no copper. The density in the copper nearest the missing part rises to 1.13e+1 A/mm against 1.48e+0 on the uncut plane. The return crowds into the copper nearest the missing part, which is where the extra resistance comes from — the current is not spread over less copper by choice but by there being none. — Fig. 6 A track three heights past the plane’s edge, at 100 MHz, with no copper beneath it. The return still uses the plane, peaking at 11.3 A/mm in the copper nearest the track and spreading much further out than the centred distribution does. The loop’s inductance is 1.82 times its centred value and its resistance 1.61.

A slot is copper taken away, and it behaves differently

An edge moves copper. A slot removes it, and the two are not the same measurement even where the gap beside the track is the same width.

A slot cut along the track and centred under it takes away exactly the copper the return most wants. Two track-heights of slot costs 1.146 times the inductance, ten heights costs 1.727 and forty heights — eight millimetres, which is a generous connector cut-out — costs 2.35. Beside that, a slot of no width comes out at 1.000000000000, which the solve returns rather than being told, and which is the check that the geometry is being described and not assumed.

At direct current a slot costs what its missing copper costs and no more: a two-millimetre slot in a fifty-millimetre plane raises the resistance to 1.0417 milliohms over a hundred millimetres, which is fifty divided by forty-eight to four figures. The inductance at direct current rises to 1.045 times, which is much less than it rises above the band, and again the two ends of the band disagree about the size of the problem.

The resistance above the band behaves as it does past an edge, and for the same reason. A two-height slot still lets the return gather close, so it crowds either side of the cut and the resistance is 2.39 times. A forty-height slot pushes the return four millimetres out on both sides, where it has plenty of room, and the resistance falls to 0.32 times the uncut plane while the inductance is at its worst. A wide slot makes the loop quieter as a resistance and much louder as an inductance, and a design that measures only one of them will read the wrong sign.

A slot two heights wide costs 15 per cent of inductance, and one forty heights wide costs 2.4 times. computed by solving, not by drawing at 100 MHz. A slot of the stated width is cut in the 50 mm plane, centred under the track and running parallel with it, and the loop's inductance and resistance are divided by their values on the uncut plane. The return goes round the slot, which is a longer way and encloses more area: two heights of slot cost 1.146 times the inductance, ten cost 1.727 and forty cost 2.35. A slot of no width is exactly one, which the solve returns rather than assumes. A slot that CROSSES the track is a different and three-dimensional question, and nothing here speaks to it. — Fig. 7 Inductance and resistance at 100 MHz against the width of a slot cut along the track and centred under it, divided by the uncut plane. The inductance rises monotonically with the slot’s width; the resistance rises to 2.39 times for a narrow slot and then falls to 0.32 times for a wide one, because a return that cannot gather does not crowd.

What the return does about a slot

The distribution shows what the two ratios are made of, and it is not the picture the word slot suggests.

A ten-height slot does not force the return to take a long detour along the board. It forces it to stand off two millimetres either side of the track, where it forms two peaks rather than one — 6.49 amperes per millimetre at a hundred megahertz, against 14.0 at an edge and 1.48 on the uncut plane.

Below the band it does something a slot is not supposed to do. At a hundred kilohertz the slotted plane’s peak density is 0.63 of the uncut plane’s, and at a kilohertz it is 0.98 — lower, not higher. An edge raises the peak at every frequency, from 1.08 times at a kilohertz to 9.5 times at a hundred megahertz, because it crowds the return into the copper left on one side. A slot runs under the track, so below the band its first effect is not to crowd anything: there is nothing gathered there to interrupt, and forbidding the gathering spreads the return rather than concentrating it. The peak rises only once the return would have been under the track, which on this stack-up is above a few megahertz.

That is the sharpest difference between the two cuts, and it is invisible in the inductance: the slot’s inductance penalty is 1.045 times at direct current and 1.727 above the band, rising monotonically throughout, while the density it produces goes the other way first. The loop those two peaks make with the track is wider than the one the single peak makes, which is the inductance, and it is no more crowded than the uncut plane’s, which is why the resistance barely moves at this width.

That also settles what a slot does to the loop’s magnetic moment, which is the quantity the corner that is three decades wide found surviving from the estimate it replaced. On an uncut plane the moment above the band is the track’s current times its height times its length, whatever the current does sideways, because the return runs in the plane’s own surface. A slot breaks the symmetry that made that true: the two peaks are not equidistant from the track in the way a continuous distribution is, and the area the loop encloses is genuinely larger. The inductance says so — 76.45 nanohenries against 44.26 for a ten-height slot — and a radiated estimate built on length times height is low by that factor rather than exact.

The return goes round a slot 10 heights wide. computed by solving, not by drawing at 100 MHz. Current per millimetre of plane against distance sideways from the point beneath the track, measured in track heights, for one ampere in the track. The heavy curve is the cut plane and the faint one the same track over a centred plane of the same area; the shaded region is where there is no copper. The density in the copper nearest the missing part rises to 6.49e+0 A/mm against 1.48e+0 on the uncut plane. Above the band a slot raises the peak, because the return would have been under the track and is not allowed to be. — Fig. 8 A slot ten heights wide, at 100 MHz. The return stands off either side of it in two peaks of 6.49 A/mm rather than one of 14.0, spread over copper that is not crowded — which is why the loop’s resistance is within two per cent of the uncut plane’s while its inductance is 1.73 times.

What the cross-section can and cannot say

Three limits belong to the method, and each of them removes a question a designer would like answered.

A cross-section has no length in it. The solve is per metre of a uniform geometry, so it describes an edge or a slot that runs along the track. A slot that crosses the track — which is what a split plane usually is, and what most of the folklore about slots is actually about — is three-dimensional, and so are the vias at either end of a real run. Nothing here speaks to either, and the numbers above should not be lent to them.

The plane is thin. The current is taken as uniform through the copper’s thickness, which holds until the skin depth reaches it; for half a milliohm a square that is thirty-four micrometres of copper and 3.68 megahertz, as the resistance that grows with frequency measures. The ratios quoted above the band are therefore the limit of this model rather than a measurement of a real plane at a gigahertz — and the inductance ratio is within a part in a thousand of its limit by a hundred megahertz, while the resistance ratio is still moving.

What an edge radiates is not in it. A solve for the currents inside the copper says where they are; it does not say what leaves the board. The crowded density beside an edge is exactly the condition an emissions engineer is worried about, and it is the one thing this cross-section cannot turn into a number.

What it can report is the pair of ratios, and they are enough to settle the rule of thumb that prompted the question. Keep three track-heights of copper beside a track and an edge is worth about a per cent of inductance; keep ten and it is worth a tenth of one. Both are ratios to the track’s height above the plane, both hold at any size, and both are invisible to a measurement made below the band — where the inductance penalty is largest.

That last clause changes one entry in the edges that are lengths. The two hundred micrometres between a track and its plane already decided where the current goes and how wide the band is. It also decides how much copper has to lie beside the track, and the answer is the same length again, three times over.

Still open: a slot that crosses, two tracks, and what leaves the board

The crossing slot, which is the one everybody means. Every number here is for copper missing alongside the track. The case that matters more — a slot the track runs over, so the return has to travel to the end of the slot and back — cannot be posed in a cross-section at all, because the geometry varies along the run. It needs a solve with two dimensions of current in the plane rather than one, and the quantity it would produce is a length rather than a ratio: how far along the slot the return detours, and therefore how much extra loop the crossing costs. The ratio measured here for a slot running along the track is the floor such a solve must sit above.

Two tracks sharing one plane, which the centred solve left open and the edge sharpens. Below the band two returns overlap however far apart the tracks are routed, and as they gather the shared fraction falls. Near an edge both returns are pushed into the same crowded copper, so the sharing should be worse there and should stay worse further up the band. The same cross-section with two tracks in it would give the shared fraction frequency by frequency, and it would turn the millivolts in the wire’s lumped common impedance into a property of a plane with a position on it.

The density, pointed at what leaves the board. The crowding beside an edge is fourteen amperes per millimetre for one ampere in the track, ten times the centred figure, and it sits in the last few hundred micrometres of copper. Turning that into a radiated field is a different calculation from the small-loop expression where the current comes back uses, because the source is a sheet current with a sharp end rather than a small loop. What can be said without it is that the two geometries whose loop inductance is nearly the same — a centred plane and one three heights wider than the track — have current densities at the edge that differ by a factor of three, so inductance is the wrong instrument for that question and this essay does not pretend otherwise.