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

Lumped-element — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 2 · limits
  2. 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 0.443 per cent at 100 ns where the staircase happens inside the edge. It reaches ten per cent at an edge of 4.25 ns and one per cent at 41.6 — which is 29.8 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 0.44 per cent on a hundred-nanosecond one, reaching one per cent at 41.6 nanoseconds. The boundary is a rise time, it is thirty times the conductor's round trip, and the repetition rate is not in it anywhere.

    part 3 · limits

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