Short Line Pi

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A line short enough that its travel time falls below the time step, so it is modelled as a lumped π section.

A 10 km line entered as per-kilometre R, L and C rather than as $Z_C$ and $\tau$. At the sample's 50 µs step the implied 33 µs travel time is too short to represent as a travelling wave, so the line is modelled as a single nominal $\pi$ section.

The line here is described the way a datasheet describes one — a length of 10 km with R' = 0.05 Ω/km, L' = 1 mH/km and C' = 0.0111 µF/km — rather than by a surge impedance and a travel time. Those numbers imply Zc = √(L'/C') ≈ 300 Ω and τ = d√(L'C') ≈ 33 µs, and the run's time step is 50 µs. A travelling-wave model needs to read conditions from one travel time ago, and one travel time ago is less than a single step, so there is nothing for it to read. NumaSim detects this before the run starts and models the line as a single nominal π section instead. It does so silently, which is part of the lesson.

With the selector on 300 Ω (matched to Zc), run it and watch Vr. It starts moving on the very first step — a π section has no delay to give — then overshoots to about 112 V and rings for a few steps before settling near 100 V. Now reduce the time step to 5 µs and re-run: the line becomes a travelling-wave model with 7 steps of delay, Vr sits at zero for 35 µs, and then steps cleanly to 100 V and stays there.

Comparing the two runs is the point of the sample. The fallback is a safety net, not a rescue: it swapped one error for another and kept going. That overshoot was never physical — it is the π section's own LC resonance, at a period of roughly 150 µs, which at a 50 µs step was being resolved with three points per cycle. A π section is the right model for a genuinely short line, not a way to tolerate a step that is too big. Used in the EMT course, Chapter 9 (losses, lumped lines, and the time step).