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

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What you need to run it: the Python API and the Machines module. Opening the model and reading results that are already there never requires a licence.

At a glance

Quantity	Value	Unit
Speed	3000	rpm
Electrical frequency	200.0	Hz
Average torque	45.903	N·m
Torque ripple	38.3	%
Cogging torque, peak-peak	0.327	N·m
Back-EMF, RMS	107.3	V
Mechanical power	14.421	kW
Efficiency	94.15	%
Power factor	0.812	—
Total losses	895.2	W
Max B, stator tooth	1.531	T
Mesh	8156 nodes, 15939 elements, P1	—
Wall time, end to end	85	s

A 15 kW traction-style motor with the magnets buried in a V under each pole. Open this one if you want to see reluctance torque doing real work — a third of the output here — or if you want to draw your own rotor rather than pick one from the list.

The rotor is a DXF. The script draws it, imports it and then places a region on each feature: lamination, two magnets, the flux barriers. Nothing is created for you, not even the rotor iron. That is the whole custom-rotor route, and it is currently the only way to get a buried magnet out of Nabla.

One pole is solved — three slots over 45°, closed anti-periodically.

The design

Layer	Radii (mm)	Set by
rotor	0.0 → 49.25	solid lamination to the axis, see below
centre bridge	0.0 → 0.75	centreBridgeHalf, the d-axis iron the barriers stop on
V vertex	34.000	vertexRadius, where the two arms meet
magnet corners	34.9 → 47.1	5 mm thick, 12 mm long, on a 110° V
barrier tips	→ 48.05	bridge, 1.2 mm under the rotor surface
air gap	49.25 → 50.0	airGap 0.75
stator bore	50.000	ID 100
slot bottom	68.000	slotDepth 18, slotWidth 6
stator OD	80.000	OD 160, back iron 12

Trapezoidal teeth, 80 mm axial length, four pole pairs, distributed full-pitch winding.

The rotor drawing

One pole, drawn about the +x axis: twenty lines and eight arcs, generated from ten numbers in the script's VPOLE dictionary. The V is specified the way a V-IPM usually is, by the opening angle between the arms measured at the vertex, opening towards the air gap. It is 110° here. The Prius uses 145° on the same pole count.

Two details follow from that angle rather than being chosen:

The end barriers run radially out to the bridge, not along the magnet axis. At 110° each arm points 55° away from the d-axis, so a barrier continuing along the magnet would hit the pole edge long before it reached the surface.

The magnets are short — 12 mm on a 38.7 mm pole arc. The arms cross the 45° pole wedge quickly, and what limits their length is the q-axis rib at one end and the 1.2 mm surface bridge at the other.

Change VPOLE and the drawing follows: corner radii, tangent points, arc angles, region seeds and magnetisation directions are all derived from those ten numbers, so changing the V angle is one edit.

The .dxf is an output, not an input. It is regenerated on every run, so editing it by hand and re-running throws the edit away. Change VPOLE instead, or pass --keep-dxf to import the drawing on disk unchanged — but read that flag's warning first, because region seeds and magnetisation directions do not follow a hand-edited drawing.

Why the rotor has no bore

The rotor is solid steel all the way to the axis, which looks wasteful and is deliberate.

The machine module draws two radial segments from the axis to close the sector, and it does not split them where your drawing happens to touch. An imported entity that ends part-way along one of those segments makes a T-junction, the polygon description cracks, and a neighbouring region floods across the crack. A bore drawn as an arc across the pole pitch is exactly that case.

The failure is silent. The model meshes, the band check passes, the solver converges, the field plot looks normal, and only the flux is wrong — 2.6 times low when this bit a real design. Keep every drawn feature clear of the sector edges, and if you need a bore, put it in as part of the same drawing rather than crossing the seams.

The script checks the mesh for the defect anyway rather than trusting the drawing: every region seed has to own elements, and a permeability probe in the lamination has to read steel rather than air.

Winding and materials

Stator and rotor core	M-19 Steel, nonlinear BH
Magnets	N35SH, chosen because it carries a demagnetisation curve
Magnet conductivity	6.7 × 10⁵ S/m (sintered NdFeB), for the eddy currents
Magnetisation	±35° from the d-axis, perpendicular to each arm
Flux barriers	air
Winding	distributed double layer, q = 1, full pitch
Turns per coil	10, one parallel branch
Slot fill	0.45
End winding	12 mm extension, resistance and inductance included

The magnet grade matters for more than temperature here. The demagnetisation-risk field needs a material that carries a demagnetisation curve, and not every magnet in the library does.

Operating point

Current	50 A peak at 30° current angle, so I_d = −25 A, I_q = 43.3 A
Speed	3000 rpm, so 200 Hz electrical
Window solved	two electrical periods, 48 steps each
Eddy currents	on, in the magnets
Mesh	1.2 mm default; 0.5 mm in the rotor core, 0.8 mm in magnets and barriers

The current angle is the point of the machine. A negative d-axis current is what turns saliency into torque; at I_d = 0 an IPM gives you its magnet torque and nothing else.

Two periods rather than one, because of the eddy currents. With eddy currents on, the transient is a genuine diffusion problem starting from zero field, so the first steps carry a startup transient with almost no torque. That is physics, not a defect, but it is not the steady state you want summarised. The summary reduces the last electrical period, so a second period is enough to leave the startup behind. Summarising the first period instead put the torque ripple at 122 % of the mean and the power balance 24 % out; at two periods they read 38 % and 0.2 %.

Running it

Open pmsm_ipm_v_shape.nbl and press Solve — about a minute. The model ships; the solved field does not, because it is a 30 MB file.

From Python:

python build_pmsm_ipm_v_shape.py             # build, solve, render, report
python build_pmsm_ipm_v_shape.py --no-solve  # geometry and mesh only, about 15 s
python build_pmsm_ipm_v_shape.py --no-eddy   # without the magnet eddy currents
python build_pmsm_ipm_v_shape.py --keep-dxf  # import the .dxf on disk as it stands

What to look at

The torque split. This is why the example exists. Put the solved inductances and magnet flux into the dq torque equation:

T = 1.5 · p · [ λ_pm · I_q + (L_d − L_q) · I_d · I_q ]
  = 29.8 (magnet) + 14.4 (reluctance) = 44.2 N·m

against 45.9 N·m from the field solution, so the dq model accounts for 96 % of it, and a third of the torque is reluctance. Compare that with E02, where a spoke rotor with a similar saliency ratio gets only 2.5 % from the reluctance term — the difference is the current angle and where the saliency sits.

One trap in that arithmetic: the inductances come back in microhenry, not henry.

Demagnetisation risk. demag_risk.png shows the field, and it reads a few per cent at the magnet centres at this operating point. The number is a percentage of the way from the working point towards the knee of the demagnetisation curve at the material's hottest published curve, so 0 % is comfortable and 100 % is the knee. Watch the corners of the magnets nearest the barriers — that is where a real IPM demagnetises first.

Saliency. L_q / L_d is 2.22. Both come from small test currents applied on top of the magnet state, so they include stator saturation, not just the rotor geometry.

The region map. region_map_mu.png colours the model by permeability, which is the quickest way to confirm that steel is steel, magnet is magnet and the barriers really are air. On a drawn rotor that is worth a look before you trust anything else.

Torque ripple is high — 38 % — and it is the design. One slot per pole per phase, a flat-topped pole and no skew leave nothing to suppress the slot harmonic. Skew is the usual answer; E03 shows what it buys.

Try this next

• openingAngleDeg, 110° towards 145° — the arms turn more tangential, the magnets can grow, and the machine trades saliency for magnet torque.
• centreBridgeHalf, 0.75 → 1.5 mm — a wider centre bridge is mechanically stronger and magnetically worse. It short-circuits more magnet flux, and back-EMF and magnet torque fall with it.
• --no-eddy — run it against the base case to see what the solid magnets cost. One period is enough without them.
• Sweep the current angle from 0° to 45° to trace this rotor's own MTPA curve. At 0° the reluctance term disappears entirely.

About these numbers

The dimensions are invented but plausible. This is not a copy of a published design and no measurement backs it. The validated V-magnet IPM is the 2004 Toyota Prius traction motor, and that is the one to cite for an IPM accuracy claim — see the validation dossier.

The rotor iron loss reads zero here. That is because a region you seed yourself starts with zero Steinmetz coefficients and the loss terms multiply by them, so switching iron loss on is not enough on its own. E06 shows how to copy the coefficients onto a drawn region, and what the answer looks like once you have.

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