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

Emissions — where it appears

The electromagnetic energy a circuit radiates or conducts onto its cables, usually measured against a regulatory limit in a stated band. On a differential link they come mostly from the common-mode current an imbalance creates, since a balanced pair's fields largely cancel.

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 Three imbalances, three slopes: a 2% amplitude imbalance is 1% common mode at every frequency, 1 mm of skew overtakes it above 455 MHz, and a 5 ps rise-time mismatch climbs at forty decibels a decade. computed by solving, not by drawing, from the transforms of the two conductors' edges and checked against the sampled edges transformed by FFT to two per cent: the common mode at the far end of a pair, as a fraction of the differential, for three separate imbalances between its halves. An amplitude imbalance of 2% gives 1.00% at every frequency. A skew of 1 mm on FR-4, 7.00 ps, gives sin(ωΔτ/2), rising at twenty decibels a decade: 2.20% at 1 GHz. Edges of 50 and 55 ps, perfectly timed and equal in size, give a common mode rising at forty decibels a decade: 0.043% at 1 GHz and 5.38% at 10 GHz. The skew passes the amplitude imbalance near 455 MHz, which is a/(2πΔτ).

The common mode that has three slopes

A differential pair turns into common mode wherever its two halves differ, and they can differ in three ways — in size, in timing and in edge rate — each with its own slope. A 2 per cent amplitude imbalance is 1 per cent common mode at every frequency. A millimetre of skew rises at twenty decibels a decade and overtakes it above 455 MHz, a/(2πΔτ). Edges of 50 and 55 picoseconds, perfectly timed, rise at forty: 0.043 per cent at a gigahertz and 5.4 at ten. In time they are a step that stays, a pulse with an area, and a doublet with none — each slope is a moment that vanishes. And an amplitude imbalance and an edge-rate mismatch of the right signs cancel at one frequency, down 74 dB, which half a picosecond of skew fills back to −25.

lines · Differential line

Named alongside it

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

Differential lineMode conversionSkewCharacteristic impedanceCommon-mode rejectionCrosstalkReturn pathRise timeTermination

All concepts