SPM-PMSM Design Calculator: Back EMF and Cogging Torque

The magnetic field is solved with a Fourier subdomain method for radial magnets, annular air gap, centered slot openings and wider winding slots. It follows the boundary conditions of Lubin, Mezani and Rezzoug (2011), attached manuscript pp. 3–7. The implementation extends the catalog's FEMM-checked open-slot template with a separately derived Dirichlet-to-Neumann elimination of each opening and slot.

With axial vector potential $A_z$, $B_r=r^{-1}\partial A_z/\partial\varphi$ and $B_\varphi=-\partial A_z/\partial r$. Laplace's equation applies in air, Poisson's equation in the current-carrying slots and magnetized annulus. Global harmonics are matched to local cosine modes by exact angular integrals. Iron surfaces impose zero tangential magnetic field. Potential and tangential field are continuous across each opening. A small identical local slot system is eliminated once. With the angle origin at slot 1, the Q-fold slot symmetry separates the global operator into independent cosine and sine Fourier blocks. Each distinct block is inverted once and cached; every electrical-cycle solution, phase-current basis and animation response reuses these same inverses.

Maxwell stress gives cogging and loaded torque. The area mean of slot potential gives phase flux linkage, and an analytic derivative gives back EMF. Phase self-inductance is the flux response to one ampere with the permanent-magnet source removed; it includes 2D slot leakage but excludes end leakage. The independent torque/power residual checks $\overline T=(p\hat I/2)\sum_j\hat\Psi_j\cos\gamma$.

For sinusoidal sizing, the radial-flux specialization of Huang et al. (1998), Eqs. 1–9, is derived with $A_{rms}=2mN_{ph}I_{rms}/(\pi D)$:

Here $B_1$ is the peak co-rotating spatial-$p$, temporal-$p$ no-load fundamental at mid-gap, $D$ is mid-gap diameter and $n$ is revolutions per second. It is extracted from the existing no-load solutions over one complete electrical period, excluding the repeated endpoint. If $B_r^{(p)}=C(\theta)\cos(p\varphi)+S(\theta)\sin(p\varphi)$, the synchronous coefficients are $B_{1c}=\langle C\cos(p\theta)+S\sin(p\theta)\rangle$ and $B_{1s}=\langle S\cos(p\theta)-C\sin(p\theta)\rangle$, and $B_1=\sqrt{B_{1c}^2+B_{1s}^2}$. The brackets denote the electrical-period mean. The spatial coefficients are evaluated analytically from the solved Fourier coefficients, without another field solve or spatial sampling grid.

This cycle projection rejects slot-modulated higher temporal harmonics and is independent of the field-plot angle. It follows the cycle-averaged power definition used by Huang et al. (1998), Eqs. 1–9. The displayed gap-field curves remain instantaneous. Efficiency is not assumed. Both length estimates hold cross-section, turns and current fixed. The sizing equation retains only the synchronous spatial fundamental; the field-based estimate also includes higher spatial winding and slot harmonics, so the two estimates can differ. Target length is zero when target power is zero.

The dense matrices are local to the numerical function. Its compact result bundle carries explicitly identified SI quantities; the named steps convert them for display. Automatic dimensional checking covers the input conversions and downstream scalar equations, while the internal numerical matrix algebra is verified by independent references and field reciprocity.

The global zero-order potential is fixed to zero. Every phase has equal positive and negative coil sides, so the net slot-current compatibility condition is satisfied. The complete phase inductance matrix is calculated with the same turn matrix for excitation and linkage, and its reciprocity is checked.

The rotor-angle control is electrical: $\theta_m=\theta_e/p$. At 45 electrical degrees per second, playback takes eight seconds per electrical period, independently of the physical speed used for voltage and power. The dense subdomain solve does not read this control. Cached PM-harmonic and per-phase-current response matrices reconstruct the instantaneous Fourier coefficients by matrix-vector multiplication. Waveform dots use linear interpolation of the angle-independent electrical-cycle samples. The field itself is evaluated from the instantaneous coefficients, so slot-induced distortion stays fixed to the stator rather than being translated as a rigid wave.

For an equally spaced, identical Q-slot stator, the bore operator is invariant under rotation by $2\pi/Q$. It couples mechanical Fourier orders only when $k\equiv\pm h\pmod Q$. The static response cache stores precisely these entries as complex cosine/sine pairs, plus each phase's current response. The independent block inverses are cached once as symmetric upper triangles. Two real entries share each complex cache element, keeping the stored arrays small without changing their values. Each inverse is checked against its own assembled boundary operator. Non-self-conjugate sine blocks are signed versions of their cosine blocks; the zero-residue and Q/2 sine blocks are cached separately. Single-slot mean-potential and current-source projections replace the large replicated slot matrices. A shared phase-current basis is solved once. Each of the four animation-cache chunks constructs only the PM-response entries it contains and copies the required part of that shared current basis. This is an exact symmetry reduction, not a truncation beyond the selected harmonic order. Cache reconstruction was cross-checked against a direct dense solve for shifted slots, off-grid rotor angles, the 200-harmonic setting and the five-phase open-slot reference.

• Infinitely permeable, unsaturated iron; two-dimensional magnetostatics; no skew or end effects.
• Radial surface magnets with uniform remanence and recoil relative permeability exactly 1. Enter remanence at operating temperature. At least four rotor poles.
• Radial slot sides, centered openings, equal slots, and a uniform imposed current density in the wider slot only. Open-slot mode puts current throughout the open slot from bore to bottom.
• Single-layer tooth coils on alternating teeth; all coils of a phase in series; balanced odd-phase windings. The phase-current angle uses each phase's computed flux fundamental as its d-axis.
• Default cross-section: Lubin Table I, with slot width 30° and opening 12°. Default 100 turns, 6 A peak and 750 W target are design choices, not published performance data. Fill factor is copper area divided by wide-slot area.
• The yoke thickness in the drawing is schematic: the ideal-iron solution does not size the iron yoke. No stator flux barriers are modeled.

Published comparisons: the semi-closed-slot reference cases read the FEM markers in Lubin Figs. 10–12 (attached PDF p. 9). Their expected values are approximate graph readings with explicit tolerances, not exact experimental measurements. The loaded case uses the paper's stated slot RMS current density of 4.6 A/mm² and checks its approximately 9 N·m average. The open-slot case uses the repository's independent FEMM 4.2 fixture for Ackermann Table 5.2 geometry with the barriers removed; it is not Ackermann's published barrier-machine result.

Lubin Table I calls δ a slot pitch and writes π/Q = 30°. For six slots the actual circumferential pitch is 60°; δ is the 30° wider-slot angle in the governing geometry. This tool uses that physical interpretation. It does not modify equations to match the table wording.

Use the 120/20 harmonic defaults for exploration; compare against 160/30 and 200/30. Cogging is sensitive to truncation and corner fields. Published FEM agreement validates these particular geometries and linear-material assumptions, not arbitrary machines. Do not infer saturation or continuous ratings from the linear results. The 8 A/mm² copper warning is an engineering screening threshold, not a universal allowable value. Short stacks need particular attention to end effects.

The phase voltage is back EMF only. Resistance drop, inductive voltage, PWM, inverter limits, iron and copper losses, magnet heating and mechanical output losses are excluded. The adjacent-phase fundamental voltage assumes the balanced phase separation 360°/m. Length scaling is valid only while the 2D cross-section and material assumptions remain applicable.

For very narrow openings, local modal convergence must be checked in addition to the global harmonic count. Validation uses the published semi-closed-slot FEM curves and the existing open-slot FEMM fixture. These references do not replace a mesh-converged finite-element check of a new design.

Short-stack screening uses the entered active length and, when target sizing is enabled, both recommended active lengths. If any is below half the mid-gap diameter, verify end effects before using either estimate as a design dimension. The L/D = 0.5 threshold is an engineering screening prompt, not a universal accuracy boundary.