Grid-free magnetostatic oracle + fix cancelling multilayer coils

[ecto]

First step of the certified-motor work: an exact field oracle to grade calc_motor's closed form against, plus a real bug it surfaced.

add_coil multilayer was electromagnetically wrong

Stacked layers reused one spiral, so series current ran outer→inner on even layers and inner→outer on odd ones through identical copper. Consecutive layers circulated opposite ways and their axial fields cancelled — a 4-layer stack made roughly one layer's field, an even stack roughly nothing.

Confirmed numerically before touching it: the new test reconstructs the current path from board data (chaining traces, following stitch vias) and found per-layer dipole moments alternating −,+,−,+.

The stack is now one continuous spiral in angle — θ marches monotonically across every layer while the radius zigzags inner↔outer, with a stitch via at each turnaround. Circulation is constant so fields add, and consecutive layers meet at the identical (r, θ) so vias align by construction.

Also fixes a pre-existing hole collision (present in the old code, not introduced here): with an integer turn count every inner turnaround landed on the same spoke, stacking two vias in one hole (HoleToHole −0.400mm). Successive turnarounds are now fanned by a stagger angle sized from via drill + hole-to-hole minimum at the binding inner radius.

Tests cover 2/4/8-layer stacks for sign consistency, ~linear moment scaling, and DRC-clean.

vcad-kernel-magnetostatic

No iron ⇒ μ = μ₀ everywhere ⇒ linear ⇒ superposition is exact. That collapses the usual FE machinery into a sum over current segments: closed-form Biot-Savart for B, closed-form ∮A·dl for flux linkage. No mesh, no iteration, no truncation boundary — and fast enough to sit inside a design search later.

Validated four independent ways:

• infinite-wire limit
• on-axis loop closed form
• off-axis loop vs vcad-kernel-particle's elliptic integrals — a different crate, written for a different purpose
• ∇×A = B by finite differences, tying the two separately-derived integrals together

Back-iron by method of images

Real "air-core" machines are only air-core in the winding region; steel back-irons close the magnetic circuit. For the examples/pcb-motor geometry that steel is worth ~2.4× in airgap flux (65 mT free space → 106 mT one plane → 155 mT full circuit), so ignoring it is not a small error. Unsaturated mild steel is μ_r ≳ 1000, so the μ→∞ mirror is accurate to ~0.1%.

Measured precondition, now documented and enforced: the two-plane image series converges only for a magnetically balanced source set.

source	depth 8	depth 16	depth 32
one loop (net moment)	2.29e-4 T	7.28e-5 T	2.21e-5 T — never settles
6 alternating poles	2.258e-2 T	2.2598e-2 T	2.2598e-2 T — converged

Real multi-pole rotors are balanced by construction, but balance_residual() enforces it and tail_fraction() reports achieved truncation rather than assuming it.

Scope, stated honestly

• Images assume flat, parallel, unbounded plates. Real back-irons are finite discs, so ID/OD fringing is unmodelled — to be quantified against the grid solver.
• No saturation; flux_density_estimate() surfaces the working iron flux so the assumption is checkable.
• Statics only — no eddy currents, skin or proximity effect.
• Filamentary self-inductance depends on the regularization radius; L is less trustworthy than Kt.

Note for the roadmap

examples/pcb-motor was never physically built — its "verified numbers" (B_gap 0.155 T, Kt 3.7 mN·m/A) are kernel-computed, not measured; only the clearances are measured. So this oracle can grade the closed form, but the third measured column needs an actual build and dyno.

Test plan

• cargo test -p vcad-kernel-magnetostatic — 14 tests
• cargo run -p vcad-kernel-magnetostatic --example image_convergence --release
• npx vitest run in packages/mcp — 934 tests, no regressions

🤖 Generated with Claude Code

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