Axial Flux Motor Design Tool

Two supported magnetic structures

The coreless branch retains Kamper, Wang and Rossouw's three-phase overlapping and type-I concentrated winding families. The iron-cored branch follows the analytical quasi-3-D procedure in Parviainen (2005), Chapter 2: one rotor, two identical stators, electrically parallel in wye, and one slot per pole and phase. Copper temperature is prescribed rather than obtained from a thermal network.

Nonlinear magnetic solution on radial planes

Each iron-cored plane represents one pole pair with six teeth. A 30-branch, 13-mesh network includes the PM/air-gap branches, nonlinear teeth, slot-opening leakage, nonlinear stator yoke and nonlinear rotor return. This is the network structure of Parviainen Fig. 2.4 extended to close the rotor return explicitly. Opposite sides use half the central rotor thickness. Magnetic flux continuity is built into the branch-to-mesh incidence matrix $C$. The solver enforces

with damped Newton iterations, analytic differential reluctances and a residual-based stopping test. It solves both no-load and loaded states at each radius. Balanced three-phase slot MMFs are sampled at the six tooth positions; the operating current is aligned with the fundamental PM back-EMF.

The stator M600-50A law uses the published Hemeida et al. fit:

The rotor S232JR curve is a piecewise-linear reading of Parviainen Fig. 1.19, attached p. 34; its high-field tail is extrapolated toward vacuum permeability. This is a material approximation, not a material certificate. The residual and differential Jacobian include the full nonlinear slopes.

Spatial field, leakage and EMF

At every radius, Carter's correction accounts for the average opening permeance. Odd spatial harmonics use the planar field solution in Parviainen Eq. (2.18), evaluated with stable negative exponentials. The magnetic-network iron MMF contribution is converted into a work-equivalent added gap separately at no load and load, implementing the saturation correction in Eqs. (2.11)–(2.12).

Leakage includes the magnet-to-rotor / magnet-to-magnet parallel paths and the zigzag factor of Eqs. (2.16)–(2.17), plus the stated radial end-leakage factor. Source discrepancy: the product printed in Eq. (2.15), attached p. 43, can exceed unity and conflicts with the factors below unity in Table 2.1, p. 44. The tool uses the passive parallel-reluctance derivation $k_\ell=[1+XY/(1+X)]^{-1}$, where $X=R_g/R_m$ and $Y=R_mP_\ell$, instead of that literal printed product. The logarithmic leakage permeances and zigzag geometry follow the same section; no coefficient is fitted to the EMF comparison.

The sinusoidal-area magnet option is an own geometric reconstruction: the covered radial area varies as a cosine with electrical angle. At radius $r$ it gives $\alpha(r)=2\cos^{-1}[(r_o^2-r^2)/(r_o^2-r_i^2)]/\pi$. The alternative sector option has constant coverage. The reconstructed magnet mass can therefore be compared independently with the published 2.1 kg.

For both structures, radial integration gives

The iron-cored full-pitch $q=1$ winding has fundamental winding factor one. Loaded saturation is included in its developed torque constant. The two parallel stators share terminal current; their EMF is not doubled. Coreless finite-width winding factors and end turns retain the Kamper equations and measured-machine comparisons.

Calculated losses and circuit quantities

Iron-cored tooth and yoke waveforms are reconstructed over one electrical cycle from the rotational symmetry of the solved pole pair. The resolved Fourier derivative is used in Parviainen Eq. (2.38):

Each plane's tooth and yoke volumes, including both stators and the lamination stacking factor, multiply their own loss densities. The M600-50A coefficients are $k_h=152$, $\sigma=4.0\times10^6\ \mathrm{S/m}$, $k_\mathrm{ex}=2.32$ and $d=0.5$ mm (Parviainen Table 5.2, p. 113). Rotor eddy loss is omitted as in §2.2.8, not replaced with a fitted allowance.

Slotted lap-winding turn length follows Eq. (2.32), hot resistance Eq. (2.31), and three-phase copper loss Eq. (2.35). Each stator's phase resistance is halved at the common terminals. Incremental d- and q-axis inductances come from the loaded network Jacobian; an explicitly assumed end-connection permeance adds the uncoupled end leakage.

Iron-cored windage uses the conservative maximum of the four smooth enclosed-disk Daily–Nece moment-coefficient correlations, with annular torque $C_M\rho\Omega^2(r_o^5-r_i^5)/2$. Bearing loss follows Parviainen Eq. (2.45) with calculated rotor mass. Stray loss is $k_\mathrm{str}P_\mathrm{shaft}$, solved algebraically without a circular dependency. The default fraction is the lower end of the thesis's 3–5% range for machines below 10 kW. The coreless branch retains user-supplied AC resistance and rotational-loss estimates. Electrical input equals gross converted power plus copper loss; shaft power subtracts all non-copper losses.

Voltage uses the entered or calculated q-axis inductance and the wye phasor relation, with the ideal linear SVPWM line-line RMS limit $V_\mathrm{dc}/\sqrt2$. Target current and current charts retain operating-point saturation and loss values.

Diameter ratio, torque density and shear

The rebuilt-winding sweep recalculates geometry and resistance; the iron-cored branch also resolves its nonlinear field at each feasible ratio. The classic comparison holds inner-radius electric and effective magnetic loading constant:

This follows Parviainen pp. 21–22 and is consistent with Huang et al.'s general sizing framework. At fixed occupied dimensions, $V=\pi D_\mathrm{env}^2L_\mathrm{env}/4$, so both ideal optima coincide.

Two active faces share torque. Their torque-equivalent uniform tangential stress is

and the local slice stress is $\Delta T_j/(4\pi r_j^2\Delta r)$. Kasinathan (2003), pp. 88–89, supplies the historical transient in-wheel comparison values.

• Coreless geometry, finite-width winding factors and end turns retain the Kamper model, with straight unskewed active conductors. Entered fundamental fields include both rotors.
• The iron-cored model supports identical parallel stators, full-pitch $q=1$ windings, uniform rectangular slots and a linear magnet recoil line. The two-dimensional planes neglect radial flux exchange.
• Copper resistivity at 20 °C is $1.724\times10^{-8}\ \Omega\,m$, temperature coefficient $0.0038\ \mathrm{K^{-1}}$, and density $8960\ \mathrm{kg/m^3}$. Rotor steel and magnet densities are assumed 7850 and 7500 kg/m³. Leads and insulation are excluded.
• Parviainen preset values from the thesis are 328/197 mm active diameters, 12 poles, 140 turns per coil (840 per stator phase), 4 mm magnets, 1.5 mm gaps, 1.05 T hot remanence and 8.4 A total terminal RMS current at 300 rpm. Copper temperature 107 °C follows Table 3.4. The 25 mm rotor support thickness and 0.92 stacking factor follow p. 74.
• Original slot and strand dimensions are not tabulated. The prototype reconstruction assumes 45 mm slot depth, 15 mm stator yokes, 9 mm slots, 2 mm openings and tooth tips, a single 1.4 mm bare conductor, and an ideal sinusoidal-area magnet outline. These values were declared before the prototype comparison; they are not fitted to its EMF or efficiency. End-connection leakage permeance is assumed 0.2.
• The prototype preset's 400 mm occupied diameter, 180 mm occupied length, 650 V bus, 2 kg shaft/hub mass and ambient-air properties are packaging/supply assumptions, not published prototype specifications. Custom mode exposes their respective design inputs.
• The in-wheel preset remains an illustrative coreless design: 400 mm active diameter, Di/Do 0.60, 32 poles, concentrated coils, 15 turns, 12 × 1 mm strands, 18 mm winding thickness, 0.65 T fundamental field, 200 rpm, 70 A RMS and 96 V DC. Its AC resistance factor 1.1 and rotational-loss estimates are assumptions.
• Shaft output and efficiency clip to zero below total drag. The unclipped power balance remains available for conservation checks. Magnet temperature is represented by the entered hot remanence; no thermal feedback or field weakening is solved.
• The occupied cylinder includes the bore. The axial stack check includes two magnets and both gaps in either structure, two rotor disks plus one winding pack for coreless machines, or one rotor disk plus both complete slotted stators for the iron-cored structure. Housing and end-turn clearance still need additional packaging allowance.

Use for low-speed preliminary electromagnetic comparison of the two supported structures. Coreless concentrated pole counts must be divisible by four; the overlapping and iron-cored $q=1$ windings require an even pole count. Parallel circuits must divide the coils per phase. Bare-strand and copper-area checks are necessary packing checks, not an insulation or manufacturing specification.

The original Kamper reference cases retain their independent measured torque and copper-mass comparisons. The new Parviainen reconstruction predicts approximately 206.2 V phase fundamental versus the measured 211 V, 2.066 kg magnets versus 2.1 kg, and 86.6% efficiency versus 89.2% at the declared rated current. Predicted shaft output is about 4.95 kW rather than being forced to 5 kW. The reference tolerances are 15 V, 0.15 kg, 3 percentage points and 0.5 kW. Unreported slot/conductor geometry, the reconstructed magnet outline, the steel curves and the uncertain original stray-loss allowance prevent this from being an exact reproduction or universal accuracy claim.

The default nonlinear calculation uses four radial planes, twelve rotor positions and three odd spatial harmonics. A doubled radial no-load calculation reports quadrature sensitivity; it does not validate the magnetic model. The iron-cored ratio sweep uses five base ratios plus exact design/optimum markers, two radial planes, twelve rotor positions and two odd harmonics. This preserves the nonlinear magnetic and loss models while keeping browser work short. It can miss narrow feasible intervals and sharp field or optimum changes between samples; it is unsuitable for accurate cogging, high-order torque ripple, PWM losses or high-frequency loss prediction. Refine magnetic planes and rotor positions when inspecting a candidate. The coreless sweep retains 61 base ratios and the entered radial slice count.

Iron-cored sweep gaps indicate impossible tooth/copper geometry. Remaining points still need voltage, temperature and manufacturing checks. The selected marker reuses the higher-resolution operating-point result; other sweep points use the documented coarse grid. Current charts and target-current estimates freeze the operating-point magnetic saturation and non-copper losses.

Iron losses use a single-valued M600-50A curve and resolved waveform derivatives; rotational hysteresis, minor loops, punching damage and thermal material-property changes are not resolved. The smooth enclosed-disk windage envelope neglects protrusions, rim drag and cooling throughflow. Its thin-gap laminar limit is retained when the spacing is below the original Daily–Nece test range. Bearing coefficient and shaft mass are estimates. Stray loss is explicitly uncertain; the prototype comparison conservatively uses 3% of shaft output. Rotor eddy and inverter-harmonic losses need independent assessment, especially above 1000 rpm or with fully open slots.

The 1/√3 optimum assumes fixed inner electric loading, effective magnetic loading and envelope dimensions. Packaging and cooling can shift a complete motor's optimum. The 18, 20 and 50 kPa in-wheel benchmarks are historical transient requirements for specific applications, not universal modern automotive or continuous coreless ratings.

No thermal equilibrium, continuous rating, demagnetization, mechanical strength, magnet retention, bearing life, wheel fit or fault response is established.