Circuit Response
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Series

Return path — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The return under 10.0 cm of track, 200 µm above the plane. computed by solving, not by drawing, as an estimate from the geometry: a low-frequency path assumed to be three track-widths wide, 83.3 mΩ, and the parallel-plate inductance µ₀h/w, 125.7 nH, which is the limit for a track much wider than its height. The estimated resistance and reactance are equal at 106 kHz. Solved across the plane instead of assumed, the return does not change path at one frequency: it gathers beneath the track across a band about three decades wide, and this corner falls inside that band. Above the band the loop is the track's length times its height, 20.0 mm², and a milliamp round it at 100 MHz radiates -1.1 dBµV/m at three metres.

    Where the current comes back

    The return current under a track spreads out at low frequency and runs directly beneath it at high. Estimated from two paths chosen in advance, the change is a corner at 106 kilohertz for any track two hundred micrometres above a half-milliohm plane, with neither the length of the track nor its width in it. Solved, it is a band three decades wide rather than a corner, and the width enters it after all — but the length is still absent and the stack-up still sets the top, which is the part of the argument a designer needs.

    part 1 · lines
  2. The return under a track gathers from 26.3 kHz to 1.42 MHz, not at one frequency. computed by solving, not by drawing, on a cross-section of a 50 mm plane cut into 120 strips, each with its resistance and its partial inductance to every other strip and to the track. The solid curve is the share of the return current inside one track-height of the point beneath the track; the second is the share inside ten heights. At direct current the return spreads evenly — 0.8 per cent within one height — and far above the band it is the image-current distribution, 48.7 per cent, which the closed form gives as 48.7. Between them it gathers across three decades: a tenth of the way by 26.3 kHz, half by 283 kHz, nine tenths by 1.42 MHz, shaded. The resistance of the path equals its reactance at 1.59 kHz, where the return has not yet moved. The single corner estimated from a path three track-widths wide and a parallel-plate inductance is 106 kHz, 28 per cent of the way through the band.

    The corner that is three decades wide

    Where the current comes back put the change in a return current's path at 106 kilohertz, from a low-frequency path assumed three track-widths wide and an inductance taken from a parallel-plate formula. Solved across a plane cut into a hundred and twenty strips, the loop's resistance equals its reactance at 1.59 kilohertz, where the current has not moved at all, and the return then gathers beneath the track over three decades — half of the way by 283 kilohertz, nine tenths by 1.42 megahertz. The single corner is a point about a quarter of the way through a band.

    part 2 · lines
  3. 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.

    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.

    part 3 · lines
  4. Two tracks 1 mm apart share all of the plane's resistance at direct current and 14.5% of it above the band. computed by solving, not by drawing, across a 50 mm plane cut into 239 strips, with two tracks 200 µm above it and 1 mm apart — 5.0 heights. One track carries the current and the voltage along the other's loop is measured. The shared resistance, as a fraction of the driven loop's own, is 1 at direct current, where both returns spread across the whole plane and share its 10 mΩ/m; it falls through a half at 183 kHz and settles at 0.1454 above 10.0 MHz, where each return has gathered under its own track. The shared inductance is 0.398 of the loop's own at direct current and 0.0341 above the band.

    Two returns in one plane

    Two tracks over one plane share the whole of its resistance at direct current, however far apart they are routed: 10 milliohms a metre on a fifty-millimetre plane, from tracks a millimetre apart or ten. The sharing ends across the same band a single return gathers over, and it ends sooner the farther apart the tracks are — through a half at 525 kilohertz for tracks three heights apart and at 9.88 kilohertz for fifty. Above the band what is left is the overlap of two image distributions, 4h²/(4h² + d²): 14.5 per cent of the resistance at a millimetre, 0.68 at five. And in the middle of the band the two loops' mutual inductance changes sign.

    part 4 · lines
  5. A 0.4 mm slot between the tracks leaves all of the resistance shared at DC and raises the in-band share from 4.03% to 4.98%. computed by solving, not by drawing, across the plane in strips. Two tracks 200 µm above a 50 mm plane and 2 mm apart, one driven and the other listened to: the share of the driven loop's resistance that the quiet loop's return also carries, against frequency, over an uncut plane (dashed) and with a slot 0.4 mm wide cut along the plane midway between the tracks (solid). The slot's two halves are one conductor, joined where the run ends, so at direct current the return still spreads across both and the whole of the plane's resistance is shared, 1.0000 of it. Above the band each return has gathered under its own track, and the share is 4.98% with the slot against 4.03% without: the slot is where the copper nearest the other track used to be.

    The slot that does not separate

    Cutting a slot in the plane between two tracks is the common response to coupling between their returns, and in a cross-section it looks as if it separates them. It does not, if the plane is one ground: 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 — 1.0000 with a 0.4 mm slot — and above the band the slot makes the sharing worse, 4.98% against 4.03%, rising to 15.7% as the slot widens to 1.6 mm, because it takes away the copper between the tracks and pushes each return towards the other. Only a split into two grounds removes the shared resistance at DC, at the price of each loop's resistance multiplied by 2.10, and it leaves the in-band coupling 29% lower, not gone: most of two tracks' coupling at 100 MHz is their own magnetic field, which the plane's copper does not carry.

    part 5 · lines
  6. A 0.6 mm guard between the tracks leaves all the resistance shared at DC and moves the in-band share from 4.03% to 3.50%. computed by solving, not by drawing, across the plane and the guard in strips. Two tracks 200 µm above a 50 mm plane and 2 mm apart, one driven and the other listened to, with a grounded trace of 0.2, 0.6, 1.2 mm centred between them at the tracks' height and joined to the plane at the ends of the run: the share of the driven loop's resistance that the quiet loop's return also carries, against frequency. The guard is part of the return net, so at direct current it changes nothing — the whole plane's resistance is still shared. Above the band a 0.6 mm guard moves the share from 4.03% to 3.50%.

    The guard that is part of the return

    Where a slot between two tracks makes their returns share more of the plane, adding copper instead — a grounded trace between them — makes them share less. But not much, and not at every width. At direct current a guard joined to the plane at the ends of the run is part of the same return and changes nothing: the whole plane's resistance is still shared. At 100 MHz a guard 1.4 mm wide in the 2 mm gap cuts the shared resistance from 4.03 to 2.55 per cent and the mutual inductance by half; a guard 0.1 mm wide raises the sharing to 4.44 per cent, since a thin trace beside both tracks carries both returns through its own small cross-section. The guard works by taking return current the quiet track's plane would have carried — 7 per cent of the neighbour's for a 0.6 mm guard — and it is bounded by exactly that.

    part 6 · lines

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