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

Distributed vs lumped — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 20 inductor-capacitor sections, against the line they are meant to be. computed by solving, not by drawing by the trapezoidal rule over 2,600 steps. The LC ladder reaches two per cent of full scale at 0.86 delays, before the wave picture says anything can have arrived, and its plateaus are wrong by up to 0.079 V. Neither is a small correction to the wave answer; they are what a network of 20 poles does when asked to be a delay.

    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.

    part 2 · lines
  2. 20 sections imitating a 1 m line: its delay is 1% long at 318 MHz and its group delay at 185 MHz, and it passes nothing above 1.32 GHz. computed by solving, not by drawing, as a chain of 20 series inductors and shunt capacitors carrying the inductance and capacitance of a metre of 50 Ω line of delay 4.83 ns, terminated in 50 Ω at both ends, beside the Bloch phase of an endless chain, 2·arcsin of ω over the cutoff, a section. The chain's cutoff is the cutoff 2/√(LₛCₛ), 1.32 GHz. Its phase delay is too long by arcsin(x)/x − 1 with x = f over that cutoff, 1% at 318 MHz (x = 0.2417); its group delay by 1/√(1 − x²) − 1, 1% at 185 MHz (x = 0.1404). Well below cutoff the solved chain's delay follows the closed form; nearer it the fifty-ohm terminations, which are not the LC ladder's own impedance there, add a ripple. Ten sections per wavelength is 414 MHz for this chain.

    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.

    part 3 · lines
  3. A line ten rises long stood in for by 20 LC sections, matched at both ends: the edge overshoots 17.5% and takes 1.66 rises to climb, where the line delivers it unchanged. computed by solving, not by drawing: an edge of 10–90% rise tr at the far end of a chain of 20 LC sections with a total delay of 10 tr, matched at both ends, marched (solid), against the same edge delayed by exactly 10 tr, which is what a line delivers (dashed). Overshoot 17.5%, rise 1.66 tr. With no reflections at either end, everything here is the chain's own dispersion: a section has a group delay that rises with frequency, so the fast end of the edge arrives late and rings behind it.

    The sections an edge needs

    A chain of LC sections rings on an ideal step however many sections it has, because a step has energy all the way to the chain's cut-off. An edge has a top frequency, and for a matched chain carrying one the section count has a clean answer: for one per cent of overshoot, 6 sections on a line one rise long, 183 on ten rises and 5,760 on a hundred, 5.8 times the length in rises to the three-halves power, within a few per cent from end to end. So sections per rise is not a constant. A section's phase error goes as the cube of frequency and the inverse square of the count, and it accumulates along the line. A 1 ns edge on a metre of cable asks for 62 sections, a 100 ps edge for 2,048, and the delay is right long before the edge is.

    part 4 · lines

All series