PM machine field: subdomain model

Field problem

Two-dimensional magnetostatics in the vector potential $A = A_z$, with $B_r = \frac{1}{r}\frac{\partial A}{\partial \varphi}$ and $B_\varphi = -\frac{\partial A}{\partial r}$:

The machine is split into regions where $A$ is a Fourier series that meets the iron boundaries by itself:

• Magnets and air gap, $r_R \le r \le r_S$ (both with $\mu_r = 1$): harmonics $\cos k\varphi$, $\sin k\varphi$; $\partial A/\partial r = 0$ on the rotor iron. The square-wave magnetisation $\mu_0 M_r = \sum B_k \cos k(\varphi-\theta)$, $k = p, 3p, 5p \dots$, adds $A = G_k\, r \sin k(\varphi-\theta)$ in the magnets, $G_k = k B_k/(k^2-1)$.
• Slot $a$, $r_S \le r \le r_{SL}$: modes $\cos \nu_n(\varphi-\varphi_a)$, $\nu_n = n\pi/\beta_{SL}$ (iron side walls), $\partial A/\partial r = 0$ on the slot bottom, uniform current density.
• Flux barrier $b$, $r_S \le r \le r_{FB}$: modes $\cos \nu_m(\varphi-\varphi_b)$, open to the air gap and to the outside; its mean mode carries $\ln r$ (flux crossing the barrier).
• Outside air, $r \ge r_{FB}$: decaying harmonics $(r_{FB}/r)^k$.

Linear system

The unknowns are the Fourier coefficients $\alpha_k, \gamma_k$ of $A$ on the bore (and of $A$ on the outer radius when there are barriers). Each region turns the potential on its boundary into $\partial A/\partial r$ there in closed form:

• air gap with magnets, per harmonic: $\partial A/\partial r = \lambda_k \alpha_k + s_k G_k(\dots)$, $\lambda_k = \frac{k}{r_S}\tanh\!\left(k \ln\frac{r_S}{r_R}\right)$;
• slot mode $n$: $\frac{\nu_n}{r_S}\tanh\!\left(\nu_n \ln\frac{r_S}{r_{SL}}\right)$; barrier mode $m$: $\coth$ and $\operatorname{csch}$ of $\nu_m \ln\frac{r_{FB}}{r_S}$ linking its two open ends.

Continuity of $A$ over every opening gives the slot and barrier modes (projections P_SL, P_FB); continuity of $H_\varphi$ along the bore — zero on the teeth — projected on $\cos k\varphi$, $\sin k\varphi$ gives the equations (field_matrix). The outside unknowns are eliminated first (Schur complement, bore_inverse). The matrix does not depend on the rotor angle or the currents, so all rotor angles are solved with one inverse.

Results

• Torque from Maxwell's stress at the mid-gap radius $R_c$; only the magnet harmonics carry it: $T = \frac{L R_c \pi}{\mu_0}\sum_k k\, G_k W_k\,(\alpha_k\cos k\theta + \gamma_k \sin k\theta)$, with $W_k$ the Wronskian of the two gap solutions.
• Flux linkage of phase $j$: $\psi_j = L \sum_s N_{sj}\,\bar A_s$, $\bar A_s$ the mean potential over slot $s$ and $N_{sj}$ the signed turns.
• Back-EMF $e = \omega\, \mathrm{d}\psi/\mathrm{d}\theta$ from the exact derivative of the magnet sources.
• Inductance: flux linkage per ampere with the magnets removed (with $\mu_r = 1$ magnets it does not depend on the rotor angle).
• Currents under load: $i_j = -\hat I \sin(p\theta - \delta_j + \gamma)$, $\delta_j$ the electrical angle of phase $j$'s no-load flux, so $\gamma = 0$ is the q-axis.

• Iron is infinitely permeable (no saturation, no iron losses); two-dimensional field (no end effects).
• Magnets are radially magnetised with recoil permeability 1; the slots carry a uniform current density; slots and barriers have radial sides; barriers cut through the stator from the bore to the outer radius, so the stator consists of separate segments, with air all around them.
• Winding: single-layer tooth coils on alternating teeth (em.windings, type "concentrated"); slot 1 is the first slot of coil 1.
• The equations are derived here with the standard subdomain method (Lubin et al. 2010); the barrier regions follow Ackermann's extension (2025, App. B). They are not copied from a program.
• Default machine: Ackermann (2025), Table 5.2. The angles of slots and barriers in that table are read as the centres of the openings: with this reading the model reproduces his cogging torque (zero near 0°, minimum −1.6 N·m near 1.5°); with them read as start angles it does not.

• Unsaturated machines; under heavy load the real torque and flux are lower.
• Accuracy against FEMM 4.2 models of the same geometry (reference cases): flux linkage, inductance, mean torque and the torque under load within 0.3 %; the air-gap flux density within 0.5 % away from slot and barrier edges. The cogging torque converges slowly (the field is singular at the corners of slots and barriers): at the defaults (200 / 15 harmonics) it is about 3 % below FEMM, converged (400 / 25 and beyond) about 1.5 % below. FEMM is converged there: halving its mesh moves the cogging torque by 0.3 %, and iron with μr 10⁶ instead of 10⁴ by 0.05 %. To check convergence, raise both numbers of harmonics together, keeping the harmonics per slot near 0.075 × the air-gap harmonics.
• Calculation time grows with the cube of the air-gap harmonics: the defaults take about 1 s in the browser, 300 harmonics about 2–3 s.
• Ackermann reports a flux-linkage amplitude of 82 mWb and about 97 N·m at 80 A for this machine, some 5 % above both this model and FEMM for the geometry as read here; his self-inductance (0.520 mH from his model, 0.535 mH from his FEA) agrees with ours (0.542 mH) and FEMM's (0.540 mH).