PMSM Efficiency Map Calculator

The motor-specific data are from SyR-e syreDefaultMotor.mat, commit 24492399a52b3450a40885285cc7d2c4d15b5276, motorModel.FluxMap_dq and IronPMLossMap_dq. The original 255×255 surfaces are bilinearly resampled to 33×33 reference-current nodes from −1000 to 0 A on d and 0 to 1000 A on q. The table source records the transformations and attribution.

For each requested shaft torque, add viscous drag torque and construct a constant-developed-torque locus. The cross-flux relation is $T_e=3p(\lambda_d i_{mq}-\lambda_q i_{md})/2$. For an arbitrary d current, bilinearly interpolate between neighboring d-current columns. Locate the q-current cell by bisection, then solve its local quadratic exactly. This preserves the requested cross-flux torque identity while allowing d current to vary continuously.

The fundamental loss-current branch is parallel to rotational EMF: $\mathbf i_{Fe}=P_{core}\,j\omega_e\boldsymbol\lambda/(1.5\omega_e^2|\boldsymbol\lambda|^2)$. Add it to magnetizing current before computing copper loss and terminal voltage. At standstill, mapped frequency-dependent losses and this branch are zero. Use $v_d=R_s i_d-\omega_e\lambda_q$, $v_q=R_s i_q+\omega_e\lambda_d$, $P_{Cu}=1.5R_s(i_d^2+i_q^2)$, and $V_{max}=uV_{DC}/\sqrt3$. Angular equations treat radians as dimensionless, explicitly normalized by the one-radian step.

The map and operating point minimize total modeled motor loss, following Ferrari Section IV-A, step 10(b). The objective includes fundamental copper, iron, PM, supplied PWM and mechanical loss. It is not a minimum-terminal-current control map. Current and voltage are simultaneous feasibility constraints.

The d-current domain is restricted by the imposed current limit. A 33-seed bracketing scan finds all detected local minima of the constrained objective; continuous golden-section searches refine their neighboring intervals and retain the best feasible point. Infeasible candidates are ranked by the larger normalized current/voltage violation, so a narrow feasible interval can be found even when none of its initial seeds is feasible. The selected point uses 26 refinements; map cells use 14. Regression and doubled-density convergence checks cover both paths.

The separate MTPA guide minimizes magnetizing current and the MTPV guide minimizes flux magnitude at each torque within the magnetizing-current circle. They ignore resistance and losses in those theoretical guide objectives; the loss-optimal selected point need not lie on MTPA. The paths are continuously refined, not snapped to d-current grid nodes.

Hysteresis uses the dataset exponent 1.05855; iron eddy and PM losses use the squared speed ratio to the 4000 rpm source loss map. These implement the map manipulation approach of Ferrari et al., DOI 10.1109/TIA.2022.3221381, equations 1–7 and 14–15; their full mesh-level iGSE/PWM procedure is not reconstructed here.

Axial/rewinding scaling follows the public SyR-e MMM_eval_mapScale.m and MMM_scale.m: reference current is multiplied by the turns ratio, flux by length times turns, torque and iron/PM loss by length, and resistance by turns squared times the active/end-turn length weighting. Reference active and end lengths are 134 and 126.7903 mm. Copper resistance uses $R(T)/R(120)=(235+T)/(235+120)$ with Celsius temperatures.

PWM adds $3R_s I_{ripple,rms}^2$ and the supplied aggregate harmonic iron/PM increment to the power balance. Harmonic currents do not enter the fundamental dq phasor or its limits. Finally, $\eta=P_{shaft}/(P_{shaft}+P_{loss})$; standstill efficiency is reported as zero. The 25×17 efficiency map writes NaN gaps for infeasible cells and for the zero-speed column and zero-shaft-torque row. These zero-output-power axes are not assigned a color. Colored cells are clipped geometrically to the piecewise-linear white torque envelope, preventing cell corners from protruding above it; an envelope gap leaves the corresponding interval blank. The torque envelope is computed from unmasked feasibility samples, so valid standstill and zero-torque limits remain available. The display torque span adapts to the current limit, using 0.65 N·m/A as a plotting margin above the largest torque/current ratio in the reference grid. Six torque bisections, each using a continuous current/voltage feasibility search, refine the boundary to one sixty-fourth of a torque cell (about 0.51 N·m at the defaults). Zero torque is tested explicitly. A speed with no feasible torque is NaN, leaving a gap in the colored-map boundary and native envelope chart; a feasible zero-torque endpoint remains zero.

The color legend spans 60–98% for detail in the useful operating region; lower efficiencies use the lowest shade. The 80, 90, 95 and 97% contours have distinct colors and line patterns, identified by the matching legend. Refinement convergence checks compare against doubled seed density and longer searches; data interpolation remains limited by the supplied FEA table.

The implementation caches source-grid columns, reuses speed-independent torque loci whenever shaft-torque samples repeat, and carries only the best d-current coordinate through each golden-section refinement. The same electrical model evaluates the search objective directly; full terminal power tables are assembled once for the selected coordinates. The envelope reuses grid-proven feasibility and searches zero torque only in otherwise empty speed columns. With zero mechanical drag, rows beyond the MTPA current limit are excluded before the map search: in positive motoring, the core-loss current increases terminal-current magnitude above the magnetizing-current magnitude. All 33 search seeds and the existing 14/18/24/26 refinements are retained. These changes preserve the objective, constraints and 25×17 sampling. Regression trials against the previous solver gave identical map feasibility, efficiencies, envelopes and selected-torque sweeps at the defaults and at low-current, low-voltage, scaled-motor and PWM/drag cases.

• Balanced three-phase steady-state motoring; dq currents and voltages are phase peaks. RMS current is peak divided by $\sqrt2$.
• Public source motor only, including saturation and cross coupling already present in its FEA maps. Magnet temperature stays at 80 °C; no PM temperature extrapolation.
• The source's tiny negative interpolated PM losses are clipped to zero; q-axis flux is set to zero on the symmetry line $i_q=0$. These are explicitly recorded in the data table.
• Stack scaling keeps cross-section and materials fixed. Rewinding keeps slot copper area fixed. End-turn resistance is retained. Radial scaling and changes of pole count are outside this model.
• The optional mechanical loss is viscous drag: its torque rises linearly with speed from the supplied value at 4000 rpm.
• PWM ripple RMS and incremental iron/PM loss are supplied assumptions or calibration values for the selected drive condition, held constant over the map. They do not include inverter switching loss or frequency-dependent winding AC resistance.
• The source flux/loss table has 31.25 A reference current spacing. Bilinear interpolation and continuous constrained searches are used between nodes; operating points are not limited to those nodes. The torque axis adapts to the drive current limit. Contours interpolate between the feasible map samples.

Use for engineering comparison and education over 0–18100 rpm, positive shaft torque and the supplied current-map domain. The product of peak-current limit and turns ratio must not exceed 1000 A. User ranges limit axial scaling to 0.5–1.5 and turns scaling to 0.7–1.3; these exploration ranges are not validated manufacturing or thermal bounds.

Efficiency belongs to the motor terminals and shaft. Inverter, battery, boost converter and transmission losses are excluded. There is no thermal-duration prediction, continuous-rating guarantee, regenerative quadrant, demagnetization test or rotor-stress assessment.

The source dataset is a different machine from the 2010 Prius. ORNL/TM-2010/253, attached file page 71 (printed 62), reports a 96% peak motor efficiency at 650 V, between about 4500 and 6500 rpm; page 72 separates motor and inverter measurements and describes short-duration high-power capability. Use it as measured context, not as a numerical validation of this source motor.

Reference cases use stored SyR-e MTPA results, directly extracted from the public MAT file with provenance and SHA256 in the attached one-page source. Small current tolerances allow for bilinear source-data interpolation, averaged FEA torque versus cross-flux torque, the added loss branch, and low-speed loss minimization instead of ideal magnetizing-current MTPA. They check the transfer against the author's independent stored calculation; they are not a new experimental validation.

Stipetić et al. 2018 is cited by DOI for the scalable saturated-model background. Its full text was not accessible during this build. Implemented scaling is limited to the separately inspectable public SyR-e axial/rewinding equations. For production decisions use motor-specific measurements or FEA, including PWM waveforms where needed. Blank plot regions identify infeasible operating points or zero shaft output power; standstill torque capability remains in the separate envelope.

The operating point and displayed map use minimum-total-loss control. This matches the stated objective of Ferrari Section IV-A, step 10(b), rather than the previous minimum-terminal-current selector. Convergence checks compare 33/65 seed scans and 14/26/34 continuous refinements; matching feasibility and specified efficiency/current/voltage tolerances are required. These numerical checks do not replace experimental validation of the motor maps.