Examples

E01 — 12-slot / 10-pole surface-PM synchronous machine

PMSM · Surface-Mounted PM · 12 slots / 10 poles · 4.956 N·m at 2000 rpm · 91.7 % efficient · 6658 nodes;

Start here. A small surface-magnet servo motor that solves in under a minute and shows the three results a PMSM is judged on: cogging torque, back-EMF and torque under load. Repeats itself at second order for comparison.

E02 — 36-slot / 6-pole spoke-type PM synchronous machine

PMSM · Spoke-Type PM · 36 slots / 6 poles · 31.534 N·m at 1500 rpm · 95.5 % efficient · 5182 nodes · 103 s

A 5 kW spoke-rotor machine and the full dq workflow that goes with it - saliency, an MTPA current angle, efficiency and loss maps, fitted iron loss and a mass breakdown.

E03 — 48-slot / 8-pole segmented-Halbach PM synchronous machine

PMSM · Halbach Array · 48 slots / 8 poles · 22.635 N·m at 1500 rpm · 97.4 % efficient · 4256 nodes · 87 s

Three construction choices you can see in the field: a segmented Halbach magnet array, stator skew solved as five axial slices, and a phase wound as four parallel branches. Includes a five-phase variant.

E04 — 24-slot / 8-pole V-shape interior-PM synchronous machine

PMSM · Custom Rotor (DXF, V-shape IPM) · 24 slots / 8 poles · 45.903 N·m at 3000 rpm · 94.2 % efficient · 8156 nodes · 85 s

A 15 kW traction-style motor with the magnets buried in a V, where a third of the torque is reluctance torque. Draws its own rotor as a DXF, and shows demagnetisation risk and magnet eddy currents.

E05 — 18-slot / 16-pole surface-PM outrunner, 45 mm across, 20 000 rpm

PMSM - Outrunner · Surface-Mounted PM · 18 slots / 16 poles · 0.387 N·m at 20000 rpm · 90.4 % efficient · 3748 nodes · 55 s

A 45 mm drone-class outrunner at 20 000 rpm, where iron loss is the larger half of the loss budget rather than a footnote. Inverted geometry, and iron loss fitted to a table that reaches 3 kHz.

E06 — 24-slot / 8-pole interior-PM outrunner on a drawn rotor, 120 mm across

PMSM - Outrunner · Custom Rotor (DXF, interior-PM in the annular band) · 24 slots / 8 poles · 8.038 N·m at 3000 rpm · 94.5 % efficient · 3585 nodes · 41 s

A 2.5 kW direct-drive outer-rotor motor with buried magnets, drawn into the annular rotor band. The two harder routes - outrunner and custom rotor - in one model.

E07 — 36-slot / 28-bar 4-pole squirrel-cage induction machine

IM · Rectangular Tooth · 36 slots / 4 poles · 4.673 N·m at 1418 rpm · 67.1 % efficient · 5717 nodes · 109 s

The first induction machine: a 28-bar squirrel cage whose rotor current is solved rather than imposed, with the T-equivalent circuit extracted and the torque, current, power-factor and efficiency curves swept against slip.

E08 — 24-slot / 18-bar 4-pole induction machine with skewed rotor bars

IM · Trapezoidal Tooth · 24 slots / 4 poles · 10.677 N·m at 1425 rpm · 70.2 % efficient · 9698 nodes · 618 s

Skewed rotor bars solved as three coupled axial slices, on a deep bar where the skin effect matters. The slowest model here, and the one that shows what skew costs and what it buys.

Magnet latch — pull-off force and the far-field boundary

Permanent magnet · U-yoke and keeper · 172 N at the closed gap · balloon BC 0.01 % off a 20× box on 84 % of the nodes · 4605 nodes · 35 s

The simplest device here and the only one whose sole source is a magnet's own remanence. A gap sweep against the classical permeance model - which it misses by 34 %, for two reasons it names - and the far-field balloon boundary condition measured against a 20× Dirichlet box.

Solenoid actuator — force and inductance over the stroke

800-turn coil · 20 mm stroke × 3 currents, 33 solves · 2.0 N at the seat · L from 62.2 to 45.1 mH · ~2650 nodes · 40 s

The axisymmetric primer: model x is the radius, the solver integrates over 2π, and there is no axial length to set. The eggshell force is cross-checked against the virtual-work derivative of the device's own inductance curve - and the example is precise about where that check works and why.

Gauss gun — a coilgun from force map to muzzle velocity

85.3 N peak at 100 A · 175 A pulse · quasi-static error 11.2 % · 25.1 m/s muzzle, 3.5 % efficient · 3 min 15 s

The largest of these: one script using the static solver, the transient solver, an external circuit and a nonlinear BH curve on the same device, then integrating the mechanics in Python and measuring the error of having done so. The turn-off timing sweep is the figure that pays for it.

Single-phase transformer — a hand-built coil on a hand-built netlist

2 kVA, 212 / 53 V at 50 Hz · saturating M-19 core · turns ratio recovered to 0.01 % · 4312 nodes · 300 steps · 1 min 55 s

The circuit-coupling reference: what you copy to drive your own windings from your own netlist. Two galvanically isolated loops tied only by the core flux - and a measured demonstration that turning saturation off does not make a BH core linear at its operating point.

Cable AC loss — skin and proximity effect in a stranded cable

Eddy currents · 37-strand honeycomb bundle · 100 A at 10 kHz · outer shell 3.6× the mean · 12642 nodes · 81 s

Thirty-seven copper strands driven identically, and at 10 kHz the outermost ring does nearly all the work while the middle of the bundle goes dark. Parametrized strand count, and the same model solved twice - three periods of transient and one frequency-domain solve - to compare the two.

Eddy-current brake — drag against speed, through a sliding band

8 poles · peak drag 864 N at 10 m/s · power balance within 1.75 % · 3393 nodes · 1 min 22 s

The motion demo, and the only one whose eddy currents come from movement rather than an AC source: no impressed current anywhere, just magnets and a sliding rail. Drag peaks and then falls as the currents start shielding the rail - and the three sliding-band rules are written out where they apply.

Induction heating — a frequency sweep that becomes a thermal transient

105 W at 10 kHz · four frequencies in 11 s · time-harmonic vs settled transient 1.25 % · equilibrium 318 °C · 20 s

Why the frequency domain exists: four operating points as four complex solves, where the settled transient of one of them costs six times the whole sweep. Then the measured induced power is handed to the thermal module - explicitly, because Nabla's thermal solve is uncoupled - and heats the billet to equilibrium.

Magnetic shield — mu-metal against aluminium, 1 Hz to 100 kHz

17 solves on one mesh · high-μ factor 250, flat across five decades · aluminium wins above 29.7 kHz · 50056 nodes · 2 min 45 s

Flux shunting and induction shielding are different physics, and putting both on one axis is the whole engineering answer. Five closed-form anchors, one mesh shared node for node between the legs, and a shielding factor that a 1 mm aluminium can takes away from mu-metal above 30 kHz.

Power-module thermal stack — junction temperature and pulsed loading

2 × 100 W · T j 53.6 °C, R th 168 mK/W · 29 K ripple at 10 % duty · 5341 nodes · 35 s

The only example in either library with no electromagnetics in it at all: nine layers, all three thermal boundary condition types, and a 10 % duty pulse train whose first peak already lands within 4 K of the final steady state. A module sized on average power is sized wrong.

PMSM

Calculate the magnetic field distribution and torque in a permanent magnet synchronous motor (PMSM).

Induction Motor

Analyze the electromagnetic performance of a squirrel-cage induction motor under various load conditions.

Eddy Current in a Cylinder

Simulate eddy currents induced in a hollow conductive cylinder by a time-varying magnetic field.

bEMF in Linear Machine

Calculate the back electromotive force (bEMF) in a linear electromagnetic machine during operation.