20 inductor-capacitor sections, against the line they are meant to be
Drawn above at its default parameters, which is almost never how an essay calls it. A placement states the numbers that essay is arguing about, so the figure a reader meets is about that argument rather than about the generator — 98% of the placements on this site pass one, and the phase that raised that number from 12% found eight captions describing a figure the page was not showing.
At those defaults the edge it states is plateaus 0.08 V out at 20 sections — the right-hand slot
of the caption strip, which on this site is never used for anything else, and which is read
back out of the drawing above rather than out of the code that wrote it.
It belongs to Lines, where a wire has a length, which is to say a change to it is a
change to lib/figures/lines.js.
It takes a slider on number of sections with 7 settings,
and every one of them has passed the same assertions as the frame above — a figure whose
circuit stops doing what its caption says at any setting stops the build.
Called by 4 essays
which is the blast radius of changing it
The staircase in time
A source driving a metre of cable does not know what is on the far end of it for 4.83 nanoseconds. What it drives into during that time is decided by the cable's characteristic impedance and nothing else — and when the far end finally answers, the answer comes back as a staircase whose limit is the resistive divider the circuit was going to be all along.
A ladder is not a line
A transmission line is usually introduced as the limit of a chain of inductors and capacitors as the number of sections goes to infinity. That is true, and it gives entirely the wrong impression of how close a finite chain gets. Forty sections still ring through every plateau by five per cent, and extrapolating the fitted convergence, reaching one per cent would need about nine hundred and sixty.
The resistor at the wrong end
A lattice diagram is read at the two ends of a line, and that is where the two respectable terminations look identical: a clean step, one delay late, at the receiver. Anyone standing halfway along a series-terminated net sees half the swing held for a full round trip, which for a logic input is not a level at all. The interval is 2(1−x) delays exactly, it is zero only at the far end, and the scheme that never has it draws sixty milliamperes for as long as the level is held.
The sections a wavelength needs
A ladder of inductors and capacitors is a line only below its own cutoff, 2/√(LC) of one section, and in the frequency domain how far below can be written down exactly: its delay is too long by arcsin(x)/x − 1 and its group delay by 1/√(1 − x²) − 1, where x is π over the number of sections per wavelength. One per cent of delay needs 13.0 sections per wavelength; one per cent of group delay, 22.4. The rule of ten per wavelength is 1.72 per cent slow in phase and 5.33 in group delay. Twenty sections imitating a metre of cable are a line to a per cent of group delay up to 185 megahertz and pass nothing at all above 1.32 gigahertz.