IPMSM Design Tool: Ld, Lq, MTPA and Field Weakening

Magnetic circuit and winding convention

This is an original reduced magnetic equivalent circuit, following the nonlinear-reluctance approach described by Lovelace (2000) and Lovelace, Jahns and Lang (2002), attached source s1, pp. 2–5 (journal pp. 646–649). It replaces the publication's 15-element q-axis network with one effective series iron path and integrates its air-gap response exactly over angle. It does not reproduce that network or inherit its reported FEM accuracy.

For series turns per phase N and winding factor kw, the peak fundamental stator MMF is F1 = kF I, where kF = 3 N kw/(π p). A sinusoidal fundamental air-gap field links a phase by ψ = Cψ B1, with Cψ = 2 N kw r l/p. The peak phase dq convention gives T = 3p(ψd iq − ψq id)/2.

The equivalent pole has two parallel magnet bars, two saturated leakage bridges and one air-gap branch. Magnet area Am uses the measured magnet axial length; gap and bridge areas use the active stack length. Solving the Norton magnet circuit gives

A rectangular pole arc has fundamental B1 = kPM (4/π) Bg sin(πα/2). The explicit kPM factor retains flux after additional end and cavity leakage. The d-axis barrier gap is gd = ge + tm Ag/(μrec Am), and Ld = Lσ + Cψ kF μ0/gd.

For the q axis, the pointwise circuit solves

H(B) is continuous and piecewise linear: its slope is 1/(μ0 μr) below Bk and 1/(μ0 μsat) above Bk. The pointwise Bg is solved algebraically in gap_q. For the fundamental, split the continuous angular integral at the moving saturation boundary θs = acos(Ik/|iq|), when |iq| exceeds Ik. The equivalent gaps are d0 = ge + lFe/(μr fq) and ds = ge + lFe/(μsat fq), and Ik = Bk fq d0/(μ0 kF). Let L0 = Lσ + Cψ kF μ0/d0, Ls = Lσ + Cψ kF μ0/ds, u = min(1, Ik/|iq|), and s = sqrt(1 − u²). Direct integration of (4/π) Bg(θ) cos θ gives
$\psi_q=i_q\left[L_0+(L_s-L_0)\frac{2}{\pi}(\arccos u-us)\right].$
Its exact derivative is
$L_{q,inc}=L_0+(L_s-L_0)\frac{2}{\pi}(\arccos u+us).$
At zero current use u = 1 by continuity. These expressions also recover the unsaturated sinusoidal fundamental exactly. The moving-boundary terms cancel when differentiating because Bg is continuous at the knee. Secant Lq = ψq/iq enters torque and steady-state voltage; the chart's incremental inductance uses the analytic derivative, not a sampled finite difference.

The independent accuracy audit integrates gap_q's local differential response over 2,048 midpoint angles at 50 and 60 A without either closed-form angular weight. It allows 0.5% numerical error for the local material-slope discontinuity. A separate reference case retains the independent review's direct-integration benchmarks, 1.334 mH and 0.958 mH, with rounding tolerance. These verify the integration of the assumed circuit, not measured machine accuracy.

Supported magnet enclosure

The inferred rotor diameter is Drot = Dbore − 2gair. A single magnet's diagonal must be shorter than Drot. The conservative equivalent layout places two bars side by side in a flat tangential row within one pole sector, with margin tb. With rotor radius R and half-width a = wm + tb, the outer face offset is yo = sqrt(R² − a²) − tb and the minimum inner face offset is yi = a cot(π/(2p)). Require tm < yo − yi. If a ≥ R there is no available space. These checks reject impossible input combinations; they do not establish shaft clearance, cavity geometry or structural strength. The drawn V-shaped pairs remain schematic.

Drive optimisation

Use vd = Rs id − ωe ψq, vq = Rs iq + ωe(ψPM + Ld id), and Vlim = Vdc/√3 for linear SVPWM. Radians are dimensionless in these electrical rate equations. The solver samples 501 nonuniform q currents, denser near zero. At each iq it solves the quadratic voltage inequality for the allowed id interval and intersects it with the current circle, id ≤ 0 and the specified negative d-current bound. Torque is linear in id at fixed iq, so the better interval endpoint is selected; the highest candidate torque is returned. The MTPA comparison removes the voltage constraint. MTPV denotes a voltage-constrained optimum with slack current and d-current bounds. These are numerical approximations to the current-vector optimisation described by Morimoto et al. (1990), DOI 10.1109/28.60058; its paywalled full text was not used.

Each envelope has 41 speed samples. Zero torque beyond an infeasible voltage region means no feasible motoring current vector was found. Copper loss is 3 Rs Ipeak²/2; converted power is Tωmechanical. Mechanical, core and inverter losses are excluded.

• Balanced sinusoidal three-phase winding, sinusoidal fundamental MMF, no cross saturation and no stator saliency.
• The d axis stays linear apart from the prescribed saturated bridge leakage. Only q-axis bulk saturation is modelled. The iron B–H law has two slopes and no hysteresis.
• Two equivalent parallel magnet bars and two saturated bridges per pole; uniform magnetisation, linear recoil line, rectangular effective pole arc. The enclosure check uses a conservative flat tangential two-bar envelope inside one pole sector. The drawing is an original schematic with exaggerated clearances, not a manufacturing section.
• ORNL Table 2.7 (s3, PDF p. 55, printed p. 46) supplies the default 161.9 mm bore, 50.8 mm stack, 0.73 mm air gap, 88 series turns and 0.077 Ω cold phase resistance. ORNL's rounded rotor/stator diameters imply a 0.75 mm gap; the separately tabulated 0.73 mm is retained. Magnet dimensions 49.3 × 17.88 × 7.16 mm come from s3, PDF pp. 54–55.
• The winding factor 0.933, Carter factor 1.10, pole arc 0.75, bridge thickness 1 mm, flux retention 0.80, remanence 1.30 T, recoil permeability 1.05, effective iron length 100 mm, area fraction 0.55, permeabilities 2000/20, knee 1.8 T and leakage 0.10 mH are explicit assumed design inputs, not a measured Prius parameter set. Fit them to your winding, material data and FEM.
• Remanence and resistance refer to the operating temperatures entered indirectly by the user; temperature dependence is not calculated. The negative d-current bound is supplied by the user and does not prove magnet survival.
• Steady-state positive-speed motoring with id ≤ 0 and iq ≥ 0; linear SVPWM with no dead time, overmodulation or semiconductor voltage drop. Current ratings are phase peak values; DC test current and RMS phase current are different quantities.

Use for preliminary IPMSM design comparisons and education. A change of effective path geometry or B–H slopes requires comparison with FEM or measured dq flux maps before choosing hardware. The Prius-sized default circuit is not a validated Prius torque model: its calculated envelope is not limited to the published 60 kW, 18-second rating.

The reference cases compare magnet volume and no-load phase RMS back-EMF only. Back-EMF readings of about 150 V at 4,000 rpm and 300 V at 8,000 rpm were read from ORNL Fig. 3.6 (s3, PDF p. 68, printed p. 59). Their ±10% acceptance bands allow graph reading and reduced-model uncertainty; they do not establish inductance or loaded torque accuracy. The comparison table also includes about 445 V at 12,000 rpm.

ORNL's motor map at 650 Vdc reaches about 96% efficiency (s3, PDF pp. 71–72). Its inverter and combined maps include 225, 500 and 650 Vdc (PDF pp. 73–75). The test description on PDF p. 71 says 230 Vdc, while the map captions say 225 Vdc; the comparison envelope uses the caption value. The prose also contains a 250 V reference where Fig. 3.17 says 225 V. These measured maps are context, not an efficiency model or validation of this tool.

Excluded: cross saturation, local bridge desaturation, irreversible demagnetisation, harmonics, cogging and torque ripple, iron/mechanical/inverter losses, end winding detail, thermal limits, rotor stress and transient control. There is no full efficiency prediction. A zero envelope segment means voltage feasibility has been lost, not safe open-circuit operation. Never use the animation speed as physical shaft speed.