Chapter 2 · Living chapter
The impedance floor of the toroidal rail is copper-dominated: capacitance sets where the floor sits, copper sets how low it goes. A verified 201-point AC sweep from 100 kHz to 1 GHz measures the rail's impedance versus frequency — every point reproducible from the downloadable netlist.
A closed-ring distribution geometry distributes current through twelve copper ring segments and eleven spokes. The rail's impedance curve descends from a capacitive region, reaches a copper-dominated floor, then rises inductively. There is no impedance peak in this model — the landmarks are floor, corner, and rise.
The sweep answers exactly one question: the rail's impedance versus frequency. Doubling the cap bank moves the corner (1.58 → 1.15 MHz) but barely moves the floor — copper owns the floor. Thermal, electromagnetic, and non-linear behavior are not claimed by this small-signal AC model.
Verified sweep
Blue ESTABLISHED evidence only — log frequency × log |Z|. Floor, corner, and inductive-rise landmarks. No live browser run: the verified pipeline is the evidence.
Scenario summary
Claim: The impedance floor of the toroidal rail is copper-dominated: capacitance sets where the floor sits, copper sets how low it goes.
Evidence: Verified 201-point AC sweep (100 kHz – 1 GHz): floor 10.57 mOhm at 1.58 MHz; 1.44186 Ohm at 100 kHz. Live variant reproduced bit-exact 2026-10-03.
Explore: The knobs are C_BANK, C_ESR, C_ESL — edit the .param lines in the live netlist and re-run locally.
Fixed: Ring/spoke copper geometry (R and L), topology, sweep range, solver regime, ngspice version.
This panel uses a small-signal AC impedance model of the toroidal rail (lumped RLC). It answers exactly one question: the rail's impedance versus frequency. Thermal, EM, and non-linear behavior are not claimed. Lumped models are trustworthy to a few hundred MHz — see Chapter 3 for the transient domain.
Floor: 10.57 mΩ @ 1.58 MHz · Start: 1.44186 Ω @ 100 kHz · Inductive rise: 35.42 Ω @ 1 GHz. The curve has no peak — floor, corner, and rise only.
Live browser simulation is not yet trustworthy in this build — four attempts failed the acceptance test (see the Gate Failure Log), so the live section is removed rather than risk wrong numbers. The verified sweep above is real: download the netlist and run it yourself, free, in under a minute.
Your local runs are your own exploration — the book's evidence is the verified pipeline, not the browser.
toroid-sweep-v1.zipPanel version: CP-LB5 · Pattern: Phase-B2 Canonical v1.3 · Last update: 2026-10-03 · Evidence bundle: toroid-sweep-v1.zip