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

Propagation velocity — where it appears

The speed at which a wave travels along a transmission line, one over the square root of the inductance times the capacitance per unit length. In a uniform dielectric it is the speed of light divided by the square root of the relative permittivity, which is what turns a measured delay into a permittivity.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

10.0 cm of track, solved as a lumped circuit and as a line. The two agree to 0.030% at 3.97 MHz, where the track is one degree long, and to 30.1% at 143 MHz, where it is a tenth of a wavelength. Above that the lumped model is not approximately right; it is describing a different object.

Kirchhoff's own frequency

The current law says the current entering a node equals the current leaving it at the same instant, which assumes the signal crosses the circuit in no time. It crosses at about two-thirds the speed of light, so the law has a frequency of its own — set by nothing but the physical size of the board.

limits · Lumped element The null is a V and not a bowl: one per cent of ratio error is 5.0e-3 of the near end. computed by solving, not by drawing. The same twelve coupled sections read at 10.0 MHz, with the ratio of the two couplings swept across the null rather than sat at one setting. The far end divided by the near end is |1 − r|/(1 + r) at every point — a straight-sided V through zero, first order in the departure with a coefficient of one half, and not a rounded minimum with a flat bottom. So there is no tolerance band: a ratio one per cent off gives 4.98e-3 of the near end and ten per cent off gives 4.76e-2, and the exchange rate between them is fixed. The upper trace is the near end over the same sweep, which moves by 11 per cent while the lower one moves through 14 decades — the two ends are the same coupling read as a sum and as a difference, which is why one of them has a zero in it and the other cannot.

How wide a null is

A far end at 3.5×10⁻¹⁹ of the drive is a statement about arithmetic until somebody asks how far the two couplings may differ before it comes back. The answer has no flat bottom in it: the far end divided by the near end is |1−r|/(1+r) exactly, so the null is a V and a ratio one per cent off returns 4.98×10⁻³ of the near end. On the axis a board is built to that is a difference of 0.0081 between the two modes' effective permittivities, out of 3.99 — two parts in a thousand, and 0.675 picoseconds of mode skew over a hundred millimetres.

lines · Crosstalk 10.0 cm of track leaves 40 per cent on a 1 ns edge. The far end of 10.0 cm of 50 Ω track, driven from 10 Ω into 200 Ω, marched as the exact staircase of reflections it is and driven by a ramp rather than by a step. The staircase converges on the resistive divider the circuit was going to be all along, to a part in 10⁹, which is the number the lumped model returns immediately and with no staircase. What is left on the edge is an overshoot: 40.0 per cent at 0.1 ns, which is the first wave's own reflection coefficient and contains no rise time at all, falling to 0.399 per cent at 100 ns where the staircase happens inside the edge. It reaches ten per cent at an edge of 4.24 ns and one per cent at 40.0 — which is 28.6 times the 1.40 ns round trip. Nothing in the measurement is a repetition rate: the boundary is a transit time, a mismatch and a rise time, and the clock the signal repeats at does not enter.

The boundary that is a rise time

Kirchhoff's own frequency is 3.97 megahertz for ten centimetres of track, and a one-kilohertz clock is four thousand times below it — which is the reading that concludes wrongly. Marched as the exact staircase of reflections it is, the same track leaves forty per cent of overshoot on a one-nanosecond edge and 0.40 per cent on a hundred-nanosecond one, reaching one per cent at 40.0 nanoseconds. The boundary is a rise time, it is twenty-nine times the conductor's round trip, and the repetition rate is not in it anywhere.

limits · Lumped element Routed along the glass weave, a length-matched pair with one conductor over a bundle and one over a gap builds 63.6 ps of skew in 20 cm — the skew nobody routed. computed by solving, not by drawing by integrating each conductor's delay along its route: a length-matched pair of 0.25 mm pitch over a glass weave of 0.5 mm pitch whose effective permittivity swings between 4.2 and 4.6 across it, positioned so that one conductor runs over a bundle and the other over the gap. Routed along the weave (dashed) the skew grows in proportion to the length, 0.318 ps a millimetre, to 63.62 ps in 200 mm. The slider rotates the route.

The skew in the glass

Match a differential pair's two lengths exactly and it can still have tens of picoseconds of skew, because a circuit board is woven glass in resin and the two conductors need not sit over the same one. Over a weave of half a millimetre whose effective permittivity swings from 4.2 to 4.6, a pair of quarter-millimetre pitch with one conductor over a bundle and the other over a gap builds 0.318 ps of skew every millimetre — 63.6 ps in 20 cm, the equivalent of 13.6 mm of routed mismatch by 30 cm, and 68 per cent common mode at 2.5 GHz. Where the pair falls across the weave is a lottery from nothing to that. Rotate the route by ten degrees and the skew never exceeds 0.146 ps, however long the pair: each conductor crosses bundle and gap in turn, and the difference averages out.

lines · Differential line With the weft counted as well as the warp, the skew a pair can reach is least at 45.0° to the warp — 128 fs — and turning further off the warp makes it worse again. computed by solving, not by drawing, to first order in the permittivity's swing, the summed bound held against an integration over 64 registrations of the pair to the cloth at six angles. A 0.25 mm pair over a cloth whose warp has a 0.5 mm pitch and whose weft has a 0.5 mm pitch, each swinging the permittivity by ±0.2 about 4.4. The warp's contribution falls as the route turns off it, as the earlier figures found; the weft's rises, because the pair's projection across the weft grows as d·sin θ while its yarns are crossed less often. The worst skew is least at 45.00°, 128.3 fs. At 10°, where the warp alone allows 291.4 fs over every registration — twice the earlier figure's 146 fs, which was for a route starting over bundle and gap — the weft adds 13.8 fs. Integrated: 5° 587.6 fs, 15° 215.1 fs, 30° 139.3 fs, 40° 126.3 fs, 60° 139.2 fs, 80° 304.9 fs. The slider is the weft's pitch.

The angle two weaves allow

Turning a differential pair a few degrees off a board's glass weave stops its skew growing with length — but a cloth has two weaves, and turning off the warp turns onto the weft. Counting both, the worst skew over every registration of the artwork to the cloth is a sum of two bounds, one falling as the route leaves the warp and one rising as it meets the weft, and it is least at an angle the cloth decides: 45° and 128 fs for two identical half-millimetre weaves and a quarter-millimetre pair, against 305 at 10°; 52° for a finer weft; 67° for a flatter one. Doing the count properly also finds the earlier bound a factor of two low, and finds the obvious answer a trap: on a square weave's exact diagonal the two weaves' product stops averaging and the skew grows again, 2.2 fs a millimetre, while a degree either side it stays at 128.

lines · Differential line The longest track a 1 ns edge from a 10 Ω driver may cross inside 1% overshoot, with its load lumped at the end and spread over 32 taps. computed by solving, not by drawing: a network of line segments marched by its characteristics, the load spread over equal taps with the last at the far end, the length bisected at each total capacitance. Lumped at the end (faint, the earlier essay's curve) the limit falls from 1.83 mm to 0.88 mm and rises to 10.9 mm at 100 pF. Spread over 32 taps (solid) the valley bottoms at 1.14 mm, and the limit is 29.5 mm at 100 pF — 2.70 times the lumped length — and 74.2 mm at 300 pF.

The load spread along the track

A hundred picofarads at the end of a track ring with its inductance as a series circuit, and a 1 ns edge from a 10 Ω driver may cross only 10.9 mm inside 1% of overshoot. The same hundred picofarads spread over 32 inputs along the track allow 29.5 mm, and 300 pF spread allow 74.2. Spread capacitance is not a load at the end; it lowers the line's own impedance to Z₀ divided by the square root of one plus the ratio of the added to the line's own capacitance, and slows it, and once the loaded line's impedance falls below the driver's — 8.2 Ω for 150 pF over 30 mm — every reflection at the source returns with the same sign and the far end climbs a staircase that cannot overshoot. The valley the lumped load has at a picofarad or two survives, shallower: 1.14 mm at its lowest, against 0.88. Two taps are no better than one; from eight the track is a loaded line.

limits · Lumped element

Named alongside it

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

Transmission lineHomogeneous dielectricLumped elementMode conversionDesign tradeoffDifferential lineElectrical lengthModel rangePermittivityReflectionsSkewCapacitive load

All concepts