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

Dispersion — where it appears

A propagation velocity that depends on frequency, so that a waveform arrives spread out rather than merely late. Its two causes on a board are the conductor's resistance at low frequency and the dielectric's own relaxation at high, and the second is forced by the loss tangent rather than chosen.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Below 910 kHz a trace is a diffusion, not a line — and its velocity goes as √f. computed by solving, not by drawing. The phase velocity of an ordinary FR-4 trace against frequency, computed from γ = √((R + jωL)(G + jωC)) with a series resistance that rises as √f above its skin-effect corner and a shunt conductance proportional to frequency. Above 910 kHz the velocity is 0.4767c and does not move, which is the number every other essay in this field uses. Below it the series resistance dominates the reactance, the line is a diffusion, and the velocity falls as the square root of frequency — measured at the 0.467 power. The characteristic impedance is not a constant down there either: 1508 Ω at a kilohertz against 50.0 Ω at ten gigahertz.

The delay that is not one number

Nine essays in this field quote a delay: a length divided by a velocity, the same for every frequency, and the edge that comes out is the edge that went in. A real trace has a series resistance, and below the frequency where the reactance overtakes it — 910 kilohertz for ordinary copper — the line is a diffusion rather than a wave, with a velocity proportional to √f. What survives is that the arrival is still exactly linear in the length. What does not is the rise time, which grows as the square of it.

lines · Dispersion A permittivity quoted as one number falls 0.645 across five decades. computed by solving, not by drawing. The real part of the relative permittivity against frequency for FR-4 (ε′ = 4.4 at 1 GHz, tanδ = 0.02), from a continuum of relaxations spread uniformly in log-frequency — the arrangement that makes the loss tangent flat. The dashed line is the single number a datasheet quotes. FR-4: 4.787 at 1 MHz and 4.142 at 100 GHz, a fall of 14.7 per cent. Nothing here is fitted: the slope is what a flat loss tangent forces.

The permittivity a loss forbids

A datasheet quotes a relative permittivity and a loss tangent as two independent numbers, and they are not two numbers. A material that dissipates has a permittivity that falls logarithmically with frequency at a rate its own loss fixes — 0.129 of permittivity a decade for FR-4, so the 4.4 quoted at a gigahertz is 4.79 at a megahertz and 4.14 at a hundred. Three hundred millimetres of track loses 62.9 picoseconds of delay between 100 MHz and 10 GHz, which a constant permittivity puts at 1.3; and the constant-permittivity model smears an edge backwards, taking 180 picoseconds to reach half height and 133 more to reach nine tenths.

lines · Dispersion At the same rise time a lossy edge draws 2.9–4.0 times more from a balanced via than a clean one, and the square law survives. computed by solving, not by drawing. The largest excursion of the reflection from a balanced via and from a 0.5 pF capacitive via against the 10–90 rise time of the edge that reaches them: behind 5 to 50 cm of lossy line, with a 20 ps edge sent in, and from clean raised-cosine edges of the same rise times on a lossless line. Behind the loss the balanced via's reflection falls as the −1.97 power of the rise time and the capacitive one's as the −0.82. Against clean edges of the same rise time the lossy edge draws 2.93 to 3.97 times as much from the balanced via and 1.07 to 1.54 times as much from the capacitive one.

The edge the loss delivers

A balanced via's reflection falls as the square of the edge's rise time, so the loss in front of it should hide it. A 20 ps edge sent down 20 cm of lossy line arrives with a 10–90 rise time of 194 ps, and the via's reflection falls to 9.1 mV. But a clean edge with that same 194 ps rise time draws only 2.29 mV from the same via. The lossy edge excites it 3.97 times as much, because loss makes an edge steep in the middle and slow at both ends, 3.07 times steeper at its centre than a raised cosine of the same 10–90 time. Across 10 to 50 cm the square law survives, an exponent of −1.97. It survives shifted: the 10–90 time overstates, by a factor of three or four, how slow the edge is to anything that reads its curvature.

lines · Reflections The delay of 50 Ω FR-4 track against frequency, with nothing dispersing, the dielectric causal, and the conductor causal as well. computed by solving, not by drawing at 108 frequencies on one track: 4.4 and a loss tangent of 0.02 at a gigahertz, 2 Ω/m of copper rising as √f above 1 MHz. Held constant (faint), the delay is flat above the resistance's corner. With the permittivity allowed to disperse (dashed), it falls 209.6 fs/mm from 100 MHz to 10 GHz. With the conductor's internal reactance put beside its resistance (solid), it falls 464.4 fs/mm over the same two decades: 7.389 ps/mm at 100 MHz, 7.094 at 1 GHz, 6.925 at 10 GHz.

The conductor made causal too

A dielectric that loses energy must disperse, and the essay before this one made FR-4's permittivity do so: the track's delay then falls 209.6 fs/mm from 100 MHz to 10 GHz. The copper has the same obligation. A conductor whose resistance rises as √f has an internal impedance (1 + j) times that rise, and putting the reactance beside the resistance more than doubles the fall, to 464.4 fs/mm — the conductor's share largest at low frequency, 418 of 584 fs/mm between 10 and 100 MHz, and the dielectric's constant at about 100 a decade. On a 30 ps edge after 300 mm the toe sharpens from 74.5 to 55.6 ps and the tail stretches from 221 to 398: the rise time is 453 ps against 295, and the tail-over-toe ratio a scope can read is 8.0 at 400 mm against 2.6.

lines · Dispersion A pair with one conductor over glass-rich laminate (4.6, tan δ 0.015) and one over resin-rich (4.2, tan δ 0.025): the skew per millimetre against frequency. computed by solving, not by drawing: each conductor as its own causal line — permittivity dispersing with its own loss tangent, copper with its internal reactance — and the skew as the difference of their delays. With both permittivities held at their 1 GHz values the skew would be 0.318 ps/mm at every frequency (level line). Dispersing, it is 0.272 ps/mm at 100 MHz, 0.319 at 1 GHz and 0.366 at 10 GHz: the resin, lossier, loses permittivity faster with frequency, and the gap between the two halves widens.

The skew that moves with frequency

A pair with one half over glass-rich laminate and one over resin-rich builds 0.318 ps of skew a millimetre when each permittivity is held at its 1 GHz value. But the two regions lose differently — tan δ 0.015 against 0.025 — so their permittivities fall at different rates, and the skew is a curve: 0.272 ps/mm at 100 MHz, 0.319 at 1 GHz, 0.366 at 10 GHz. A deskew set at one frequency leaves ±2.35 ps in 50 mm at the other two. The resin half also loses 3.7 dB more by 10 GHz over 200 mm, so its edge arrives 63.7 ps early and rises in 257 ps against 202; the halves differ in timing and in edge rate at once, and leave a common-mode pulse of 151 mV for each volt.

lines · Dispersion

Named alongside it

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

Rise timeLine lossModel rangePropagation constantSkin effectCharacteristic impedancePermittivityDesign tradeoffDifferential lineLoss tangentMode conversionReflection coefficient

All concepts