SynRM Design Tool — Flux Barriers, dq Torque and Ripple

Circular-barrier sizing

Geometry and magnetic equations follow SyR-e's `xbPlane_analyticalDesign.m`, pinned at commit 24492399a52b3450a40885285cc7d2c4d15b5276, and `staircaseAnyAlpha.m`. This implementation uses a selectable total air-thickness budget instead of SyR-e's full stator/thermal design plane.

For rotor radius $r$, pole pairs $p$ and electrical q-axis tip angles $\alpha_k$, circular barrier centres lie at $x_c=r/\cos(\pi/(2p))$. The barrier radius is $\rho_k=(x_c^2+r^2-2x_cr\cos(\alpha_k/p))^{1/2}$; the half arc length is $s_k=\rho_k\beta_k$. Thickness is distributed as $h_k\propto1$, $s_k$, or $\Delta f_k\sqrt{s_k}$, normalized to the entered thickness sum. The latter minimizes the flow-through term at fixed tip positions and total thickness.

The normalized cosine MMF is averaged over each q-axis region; the region next to the d axis is set to zero. With region widths $\Delta a_k$ in radians:

Lengths in the magnetic equations are SI. Half-lengths are measured at nominal rotor intersections; rib-trimmed outlines are used in the drawing. Clearances are normal distances between concentric circles, plus shaft and pole-edge checks.

dq operating point

$i_d=I_{pk}\cos\beta$, $i_q=I_{pk}\sin\beta$. Known mode defines $\psi_d=L_di_d-L_xi_q$, $\psi_q=L_qi_q-L_xi_d$; $L_x=0$ in geometry mode. Torque is $T=\tfrac32p(\psi_di_q-\psi_qi_d)$. The matrix must be positive definite. No PM flux is included.

For total uniform electrical skew span $\sigma=p\,\sigma_m$, averaging the frozen dq matrix multiplies its anisotropic and coupling terms by $K_0=\sin\sigma/\sigma$. The common inductance is unchanged. This is a derived axial average, not a skewed FEA result. Voltage follows $v_d=R_si_d-\omega_e\psi_q$, $v_q=R_si_q+\omega_e\psi_d$; line RMS is $\sqrt{3/2}\,|v_{dq}|$. Electromagnetic power is $T\omega_m$; copper loss is $\tfrac32R_sI_{pk}^2$.

Entered-spectrum ripple

The user supplies peak harmonic amplitudes $a_6,a_{12}$ as fractions of unskewed mean torque. The reconstructed waveform is

where $K_h=\sin(h\sigma/2)/(h\sigma/2)$ and the zero-span limit is 1. This attenuation follows direct integration along the skew. Peak-to-peak ripple is $(T_{max}-T_{min})/|\bar T|$ from a 0.25° electrical grid; each modeled cosine has a sample within 0.125° electrical of an extremum, giving less than 0.04% error relative to that harmonic's peak amplitude. Stator slot count and barrier angles do not infer the harmonic inputs.

Benchmark convention

Zhang et al., EVS34, attached source s1, p. 7 Table 3 supplies independent FEA torque values and local inductances. Its numerical coupling torques follow $+\tfrac32pL_xI^2\cos2\beta$, consistent with the negative off-diagonal convention here. The printed $L_{dq}(i_q^2-i_d^2)$ term on p. 6 has the opposite sign to its cosine expression and Table 3; this tool preserves the table-consistent physics. The p. 9 1.98 N·m example is also inconsistent with its p=2 and reported inductances (which give 0.99 N·m using the stated peak-current equation); it is not used as a reference.

Balanced sinusoidal three-phase currents; amplitude-invariant Park transform; no permanent magnets. Phase current and dq values are peak quantities, while the line voltage result is RMS.

The geometry model represents a circular, symmetric, transversely laminated rotor with 4–12 poles (2–6 pole pairs). Barrier tips are evenly spaced between the entered endpoints; barrier count is 1–5. A nonmagnetic shaft and negligible end effects are assumed. Tangential ribs are shown; additional radial ribs enter only through an equivalent saturated leakage width.

The 2 T rib flux density is the SyR-e default. Winding factor, Carter factor, saturation factor and leakage inductance are user assumptions until calibrated. The saturated-rib model is valid only once the ribs are substantially saturated; it is not an unsaturated low-current model.

The selected local secant matrix is frozen for the current-angle sweep and skew calculation. There is no steel B-H iteration, cross-saturation map, temperature model or current controller. The default 3% and 1% harmonic peaks and 30° relative phase are illustrative screening inputs, explicitly editable; they are not a geometry-based ripple prediction. Default geometry is a generic 100 mm rotor, not a reproduction of a published motor.

The rotor outline omits fillets, radial rib detail, stator slots and manufacturing tolerances. The dq arrows use electrical angle; the physical rotor rotates by electrical angle divided by pole pairs. Playback is illustrative and independent of shaft-speed input.

Use for preliminary circular flux-barrier sizing and local dq operating-point comparisons. Geometry mode requires ordered tip angles, barriers inside their pole sectors, sufficient shaft and carrier clearance, and positive saliency. If the rib approximation overwhelms the d-axis inductance, increase current or use a measured/FEA magnetic model.

Known mode requires $L_d>L_q>0$ and $L_dL_q>L_x^2$. Use secant inductances obtained with the same winding, peak-current convention and operating point. The frozen-model best angle is not a validated MTPA setting for a saturated motor.

Electrical skew span must be below 90°. Uniform continuous skew and constant current distribution are assumed; step skew, axial/end effects and skew-dependent saturation are excluded. Ripple results apply only to the entered 6th/12th spectrum; other harmonics, PWM and transient torque are excluded. Smooth mode does not evaluate ripple.

Electromagnetic power is before mechanical and iron losses; no shaft efficiency is claimed. Voltage utilization above 100% requires lower speed/current, higher DC voltage or a separate field-weakening solution; overmodulation is excluded. PF is undefined at zero voltage (the numerical helper returns zero).

Reference cases validate the local dq torque calculation against published FEA, not the accuracy of the geometry estimate, rib saturation, or arbitrary harmonic spectra. Ragazzo et al. explicitly proceed from equation-based sizing to magnetic, structural and thermal FEA. This tool does not certify rotor integrity, thermal rating, vibration, acoustic performance or a safe operating speed.