Circuit Response
every model has an edge, and the figure carries it
Concept

Return path — where it appears

The route a signal's current takes back to its source, usually through a ground plane beneath the track. At high frequency it follows the track to minimise inductance; at low frequency it spreads to minimise resistance, and returns that share a path share an impedance.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

What 1.5 mm of length mismatch does to a differential pair. computed by solving, not by drawing. Two lines of the same impedance and different lengths, driven differentially. The solid rising curve is what arrives as common mode; the dashed one beside it is sin(ωΔτ/2), which is what a lossless pair gives and is the same curve until the null. The flat curve at the top is the differential signal, and it is the point: at 3.04 GHz a tenth of the launched amplitude is common mode and the differential has lost 5011 parts per million of itself. The conversion is first order in the skew and the loss is second order, so the error is not missing from the signal — which is why a pair can pass its own eye and fail an emissions test. At 95.3 GHz the closed form has a null and the real pair does not: the longer conductor is also the lossier one, and an amplitude imbalance has no null in it.

The millimetre that becomes common mode

A pair carries two modes rather than two signals, and a length mismatch between its halves converts one into the other. A millimetre and a half of skew is ten picoseconds, a tenth of the signal is common mode by three gigahertz, and the differential signal has lost five thousand parts per million of itself getting there — so the error is not missing from the signal, which is why a pair can pass its own eye and fail an emissions test. The product in the answer is ωΔτ, which is what the instruments field's rejection corner is one over.

lines · Differential line 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.

lines · Return path

Named alongside it

The objects these essays reach for when they reach for this one.

CrosstalkCharacteristic impedanceCommon-impedanceDifferential lineEmissionsGround planeGuard traceMode conversionMutual inductanceSkewSkin effectTermination

All concepts