2.5D annular model and efficiency map¶
The annular model splits the disk machine into radial slices and evaluates the field, torque contribution, yoke loading, and losses per slice. Because torque and back-EMF derive from the same summed flux linkage, the model agrees with the Layer-1 analytical model to machine precision for a perfect machine — the extra fidelity appears only where physics is genuinely radius-dependent.
import dataclasses
from pathlib import Path
from axfluxmdo import AxialFluxMotor, OperatingPoint
from axfluxmdo.models import AnalyticalModel, AnnularModel, compute_efficiency_map
from axfluxmdo.viz import plot_efficiency_map, plot_radial_profiles
OUTPUT_DIR = Path(__file__).parent / "output" if "__file__" in globals() else Path("output")
OUTPUT_DIR.mkdir(exist_ok=True)
motor = AxialFluxMotor(
outer_radius=0.08,
inner_radius=0.025,
air_gap=0.0008,
pole_pairs=14,
phases=3,
turns_per_phase=24,
fill_factor=0.45,
magnet_thickness=0.004,
back_iron_thickness=0.006,
)
op = OperatingPoint(speed_rpm=500, current_rms=25, dc_bus_voltage=48)
Layer-1 / Layer-2 parity¶
For a perfect axisymmetric machine the two layers agree exactly — a strong correctness check on both.
analytical = AnalyticalModel().evaluate(motor, op)
annular = AnnularModel(n_slices=32).evaluate(motor, op)
print(f"analytical torque: {analytical.torque_nm:.6f} N·m")
print(f"annular torque: {annular.torque_nm:.6f} N·m")
print(f"relative diff: {abs(annular.torque_nm / analytical.torque_nm - 1):.2e}")
analytical torque: 8.629426 N·m annular torque: 8.629426 N·m relative diff: 0.00e+00
Radial profiles¶
Where the fidelity shows up: yoke flux density grows with radius (pole pitch τ_p = πr/p widens outward), so saturation binds at the outer radius first — the Layer-1 mean-radius proxy underestimates it.
fig = plot_radial_profiles(annular)
fig.savefig(OUTPUT_DIR / "03_radial_profiles.png", dpi=150, bbox_inches="tight")
fig
Pole-pitch effect on yoke loading¶
Fewer pole pairs → wider poles → more flux per pole returning through the same yoke → saturation. This is why the p = 4 variant was infeasible in example 02.
for p in (8, 14, 20):
r = AnnularModel(n_slices=32).evaluate(dataclasses.replace(motor, pole_pairs=p), op)
print(
f"p={p:3d} B_yoke at r_i={r.slice_yoke_b_t[0]:.2f} T, "
f"at r_o={r.slice_yoke_b_t[-1]:.2f} T (limit 1.6 T)"
)
p= 8 B_yoke at r_i=0.58 T, at r_o=1.77 T (limit 1.6 T) p= 14 B_yoke at r_i=0.33 T, at r_o=1.01 T (limit 1.6 T) p= 20 B_yoke at r_i=0.23 T, at r_o=0.71 T (limit 1.6 T)
Wedge vs rectangular magnets¶
Rectangular magnets (constant width) over-cover the pole at small radii (clipped at full coverage) and under-cover at large radii, where most of the torque is produced — so they trade a few percent of torque for simpler magnet procurement.
rect = AnnularModel(n_slices=32).evaluate(
dataclasses.replace(motor, magnet_shape="rectangular"), op
)
delta = rect.torque_nm / annular.torque_nm - 1
print(f"wedge torque: {annular.torque_nm:.3f} N·m")
print(f"rectangular torque: {rect.torque_nm:.3f} N·m ({delta:+.1%})")
wedge torque: 8.629 N·m rectangular torque: 8.102 N·m (-6.1%)
Efficiency map over the speed–torque plane¶
Constraint-aware: grey cells violate at least one limit (thermal, current density, voltage, electrical frequency). The map takes ~1600 model evaluations and runs in well under a second.
emap = compute_efficiency_map(motor, op, max_speed_rpm=3000, max_torque_nm=12)
fig = plot_efficiency_map(emap)
fig.savefig(OUTPUT_DIR / "03_efficiency_map.png", dpi=150, bbox_inches="tight")
fig
