Chapter 4 · Living chapter
A space-filling ground plane and the falsifiable current-divider measurement that tests it. An AC sweep from 100 kHz to 1 GHz, with every point reproducible from the downloadable netlist.
A Hilbert curve is a space-filling fractal — a continuous line that visits every quadrant of a square region. As a ground-plane return path, it offers a geometry that distributes return current across the entire plane area, rather than concentrating it under a single trace. The question is not whether the Hilbert path "works" — it does, in the sense that current will flow through it. The question is how much, and at what frequencies.
Two return paths run in parallel from the load to the source: a conventional solid plane (Rp, Lp) and the Hilbert trace (Rh, Lh). A 1 A AC stimulus splits between them. The share of return current flowing through the Hilbert path is a function of frequency, set by the impedance ratio of the two paths. At DC, the plane's near-zero resistance dominates: the Hilbert share is tiny. At high frequency, inductance takes over, and the share rises toward LP/(LP+LH). The test is falsifiable: if the Hilbert path carried no meaningful current at any frequency, the share would be zero everywhere.
A 201-point AC sweep from 100 kHz to 1 GHz (.AC DEC 50) measures the Hilbert return-current share across four decades. The sweep is small-signal and linear — no transient, no non-linear effects. The model answers exactly one question: how does return current divide between the two paths as a function of frequency?
At the default values (RH = 0.39 Ω, LH = 48 nH, LP = 0.2 nH), the Hilbert share starts at 0.041% at 100 kHz and plateaus at 0.415% at 1 GHz. The low-frequency share matches the resistance ratio RP/(RP+RH) = 0.026% — the small discrepancy is the inductive contribution at 100 kHz, which is not quite zero. The high-frequency plateau matches the inductance ratio LP/(LP+LH) = 0.415% exactly. The curve is a rising sigmoid on a log-log scale, transitioning from resistance-dominated to inductance-dominated around the frequency where ω·LH ≈ RH.
The Hilbert path carries a small but non-zero share of return current, rising with frequency. It is not a replacement for a solid plane — at these default values, the plane carries over 99.5% of the current at all frequencies. But the divider test is falsifiable: change RH, LH, or LP, and the share changes predictably. Lower RH and the low-frequency share rises. Raise LH and the plateau falls. Raise LP and the plateau climbs. The geometry sets the impedance; the impedance sets the share.
Try it yourself
Scenario summary
Claim: The fraction of return current flowing through the Hilbert path varies with RH, LH, and LP.
Evidence: Verified 201-point AC sweep (100 kHz – 1 GHz), reproduced bit-exact 2026-10-02.
Explore: RH, LH, LP (safe ranges only).
Fixed: Topology, sweep range, solver regime, ngspice version.
Pre-computed from the verified 201-point AC sweep. Reproduced bit-exact 2026-10-02.
Panel version: CP-LB4 · Pattern: Phase-B2 Canonical v1.1 · Last update: 2026-10-02 · Evidence bundle: divider-sweep-v1.zip
Live experiment
Edit RH, LH, or LP and re-run the AC sweep in your browser. Your trace appears in grey dashed over the blue ESTABLISHED baseline.
the low-frequency share is set by the resistance ratio RP/(RP+RH)
the high-frequency plateau is set by the inductance ratio LP/(LP+LH)
the plane side of both ratios
Model integrity: this panel uses a small-signal AC model. It answers exactly one question: frequency-domain return-current distribution. Non-linear or transient behavior is not claimed. This is a frequency-domain model: it does not predict transient droop or ringing (see Chapter 3 for that).
Simulation settings
Model: DIV_v1. Solver regime: AC small-signal, 201 points, 100 kHz – 1 GHz (.AC DEC 50), ngspice defaults — same as the verified runs. Timeout: 10 s. No transient fallback ever.
This is a small-signal AC model. It does not predict transient droop or ringing (see Chapter 3 for that).
Reader experiments are local-only. No data is sent to Continuum Press. No reader runs are ever promoted to evidence.
Panel version: CP-LB4 · Pattern: Phase-B2 Canonical v1.1 · Last update: 2026-10-02 · Evidence bundle: divider-sweep-v1.zip