IPMSM MTPA, Field Weakening and MTPV Current Trajectory

The steady-state amplitude-invariant dq model is

The current circle is $i_d^2+i_q^2\le I_{max}^2$ and the exact resistance-inclusive voltage ellipse is $v_d^2+v_q^2\le V_{max}^2$. These equations follow Sanz et al. (2019), §§2.1–2.3, and Eldeeb et al. (2017), §§II–III. The numerical solution and drawing are original.

For $\Delta L=L_q-L_d$, the stable closed form

includes the equal-inductance limit without dividing by saliency. At standstill, $I=\min(I_{max},V_{max}/R_s)$; the d-axis ceiling clips the optimal angle if necessary. Base speed solves the voltage quadratic at this low-speed current pair.

At each speed, the $i_q=0$ section of the voltage ellipse provides the feasible d-current interval. For each candidate id, a quadratic gives the positive voltage-limited iq; the current circle supplies the other iq bound. The smaller bound is used, and 38 golden-section iterations maximize torque over id. The positive torque objective is log-concave over this interval. This solves field weakening and MTPV without imposing constant power or assuming every motor reaches MTPV. Constraint re-entry is retained. Entry speed is refined by bisection with a 10⁻⁵ relative current-slack criterion.

Maximum electrical speed is found on iq = 0 by maximizing $\sqrt{V_{max}^2-R_s^2 i_d^2}/|\psi_m+L_di_d|$ over the permitted negative-id interval. If flux cancellation is accessible and $V_{max}\ge R_s\psi_m/L_d$, no finite electrical cutoff exists. With zero resistance and incomplete cancellation, this reduces to $\omega_{e,max}=V_{max}/(\psi_m-L_dD)$, $D=\min(I_{max},I_{dmax})$.

Balanced three-phase sinusoidal IPMSM; positive mechanical speed and positive motoring torque; Lq ≥ Ld, including the equal-inductance limit. Constant phase Ld, Lq, magnet flux and operating-temperature resistance. The d-axis aligns with magnet flux; id ≤ 0 and iq ≥ 0. Current and voltage are fundamental phase peaks. The independent current and negative-id ceilings are supplied by the user.

Steady-state maximum-torque envelope only. Excludes saturation, cross-coupling inductance, harmonics, iron losses, inverter nonlinearities, bus power limits, thermal duty, friction, windage and transient voltage headroom. Use the available fundamental voltage after reserving control margin. Converted power is before mechanical losses. Flux cancellation can remove the ideal electrical speed limit; mechanical, bearing and inverter-frequency limits still apply. The finite cutoff is a zero-torque limiting speed, not a useful loaded operating point.

The envelope is solved at 301 speeds; no interpolated solution is used for the animated current command. The voltage ellipse has 721 drawing samples. Sharp constraint transitions are physical. The diagram shows the upper, motoring half of the current circle in coordinates normalized by Imax. MTPV entry is the first region with both current ceilings slack in the sampled speed span and is numerically refined; a narrower region than one speed interval can be missed by the entry marker, although the selected-speed optimization remains valid.

Sanz motor A gives 1.483 N·m against the listed 1.48 N·m nominal rating. At 1800 rpm and a 6 V bus, this constant-parameter model gives 0.573 N·m against the paper's reported 0.56 N·m simulation. Motor B gives 3.358 N·m against a listed nominal 3.3 N·m. Checks explicitly allow these small model/rating differences; the equations are not fitted to the reported values.