Cogging Torque: Slot–Pole, Magnet Width and Skew

Slot–pole symmetry follows Zhu & Howe (2000), equations (1)–(4), open repository PDF pp. 3–5: $N_c=\operatorname{lcm}(Q,P)$, selected mechanical order $\nu=hN_c$, and qualitative indicator $C_T=QP/N_c$. $C_T$ is not used to scale torque. Slot, pole and cogging pitches are $360^\circ/Q$, $360^\circ/P$ and $360^\circ/N_c$.

For a rectangular magnet-energy window with coverage $\alpha$, its Fourier numerator at the interacting order is $G_h=|\sin(\pi h r\alpha)|$, where $r=N_c/P$. This ideal trend is derived here from the window integral. Interior fundamental zeros occur at $\alpha=j/r$, $j=1,\ldots,r-1$; the widest is $1-1/r$. If $r=1$, there is no interior fundamental null. Arc and chord widths are $R\beta_m$ (angle in radians) and $2R\sin(\beta_m/2)$. The comparison field coefficient is $\sin(\pi\alpha/2)$ at fixed rectangular field plateau.

Continuous axial averaging gives $K_s=\sin(x)/x$, $x=\nu\beta_s/2$, with $K_s(0)=1$ and angles converted to radians. This is Zhu & Howe equation (2). $A_{sk}=A_0|K_s|$, and one sinusoid has peak-to-peak torque $2A_{sk}$. Waveform phase is referenced to the stack midplane, so the signed factor is retained. Numerical waveform values are rounded at ten decimal places in the sine coefficient to remove floating-point noise at exact zeros.

The same averaging at mechanical orders $P/2$ and $3P/2$ gives fundamental and third-electrical-harmonic back-EMF retention. Scuiller et al. (2020), attached manuscript pp. 6–7, independently report skew coefficients used in the reference checks. No segmentation model is imported from that paper.

• Identical equally spaced stator slots and an even total number of alternating rotor poles. Surface-PM geometry with one uniformly covered magnet arc per pole.
• The width curve assumes rectangular magnet-energy and field windows, no fringing, and unchanged field plateau. It is a normalized coefficient trend; it is not an absolute cogging torque solution.
• The supplied unskewed amplitude belongs to the selected harmonic, slot–pole pair and magnet width. Update it whenever geometry or harmonic selection changes. The 0.2 N·m default is illustrative.
• Uniform continuous skew and identical uncoupled axial slices. Signed skew sets handedness; amplitude retention is even in skew.
• The rotor-angle origin is illustrative. Only one torque harmonic is animated. The axial sketch enlarges skew and is not a manufacturing drawing.

Use for preliminary symmetry, pole-arc and continuous-skew screening in ideal radial surface-PM machines. Magnet coverage is limited to 0.05–0.99; the full-pitch boundary is excluded because neglecting fringing would create a misleading zero. The widest ideal null is not a final magnet-width recommendation: Zhu & Howe show that fringing shifts the optimum.

A theoretical zero does not imply zero measured cogging. Manufacturing variation, eccentricity, unequal magnets, slot geometry and three-dimensional fields can introduce residual and additional harmonics. A selected higher-harmonic skew zero can leave lower harmonics. Retention is not globally monotonic with skew because sinc has sidelobes and phase reversals.

The torque result only attenuates an entered unskewed harmonic amplitude. The width trend is deliberately not multiplied into that measured/FEM baseline. Neither the slot–pole indicator nor the useful-field coefficients predict total torque, efficiency, speed ripple or vibration. Stepped/skewed segments, unequal axial lengths, saturation, end leakage, loaded torque ripple, winding feasibility and mechanical design require separate analysis.