Cogging torque: causes, slot-pole selection, and reduction

Spin the shaft of a permanent-magnet motor with no current flowing and you will feel a subtle, rhythmic resistance — the motor wants to snap to certain positions. That resistance is cogging torque. Choosing the right slot-pole combination can reduce it by a factor of five or more before touching a single geometric parameter.

What is cogging torque?

Cogging torque (also called detent torque or reluctance torque) is the torque produced by a permanent-magnet motor with zero stator current. It arises entirely from the magnetic geometry of the rotor–stator interface, requiring no electrical excitation. When you rotate the shaft by hand, you feel it as a repeating "notchy" sensation — the rotor prefers discrete angular positions and resists being moved away from them.

The stator is built from laminated electrical steel — a soft magnetic material with a relative permeability μ_r of roughly 5,000 to 10,000. This high permeability means the steel offers almost no resistance to magnetic flux. The air gap, by contrast, has μ_r = 1. The slot openings, filled with air and conductor insulation, behave magnetically like an extension of the air gap. This stark contrast between the permeable steel teeth and the non-permeable slot openings is what creates the reluctance variation that drives cogging torque.

Cogging torque is distinct from torque ripple, which appears under electrical loading. Cogging torque exists at standstill and at any speed; torque ripple is the modulation of electromagnetic torque caused by winding harmonics and back-EMF shape under current excitation. Both matter for smooth operation, but they respond to different design choices.

In servo drives and precision positioning systems, cogging torque introduces a position-dependent disturbance that the control loop must reject. In consumer appliances and e-bikes, it shows up as audible noise and vibration at low speeds. In direct-drive hub motors, it can cause a "cogging" sensation felt by the rider at slow cadence. Reducing it at the design stage is far cheaper than filtering it in firmware.

Which motors have cogging torque?

Cogging torque is present in any motor that combines permanent magnets with a slotted stator. It is absent wherever one of those two elements is missing.

The reluctance mechanism

A magnetic circuit prefers the path of lowest reluctance. The stator teeth are made of laminated electrical steel, which is highly permeable (μ_r ≈ 5,000–10,000). The slot openings between teeth are filled with air and conductor insulation, both of which have μ_r = 1 — the same as free space. This means the steel teeth are magnetically transparent to the rotor flux, while the slot openings act as extensions of the air gap, presenting much higher reluctance.

As the rotor turns, the reluctance seen by each magnet cycles between a minimum (magnet over tooth) and a maximum (magnet over slot opening). When a magnet face is aligned directly over a tooth, the flux crosses only the thin mechanical air gap through highly permeable steel — low reluctance, low stored energy. When the magnet face is over a slot opening, the flux must span a wider effective gap through air — higher reluctance, higher stored energy.

The rotor magnet is attracted toward positions of minimum energy — that is, toward the stator teeth. Moving away from a tooth requires doing work against this attractive force. The cogging torque is the rate of change of stored magnetic energy with rotor angle:

The diagram below shows a developed (unrolled) view of the stator–rotor interface. On the left, the magnet is centred over T2 — equal overlap on both sides, minimum reluctance, zero net force, stable equilibrium. On the right, the magnet has shifted so that it overlaps much more with T3 than with T2. T3 now presents a shorter, lower-reluctance flux path than T2. The magnet is pulled in the direction of greater overlap — toward T3. At exact symmetry (magnet centred in the slot) there would be no net force either; it is the asymmetric overlap that creates the cogging force.

The periodic variation of stored energy with rotor angle drives the cogging torque. The spatial frequency of this variation — how many times it cycles per revolution — is determined by the slot-pole combination.

LCM(Q, P) — the key metric

The number of cogging torque cycles per mechanical revolution equals the Least Common Multiple of the slot count Q and the pole count P:

Each cogging cycle spans an angular step of:

The LCM can be computed from the greatest common divisor:

The LCM counts the total number of rotor positions per revolution at which a magnet edge and a slot edge are simultaneously coincident — call these alignment events. Each alignment event is a moment of peak cogging force for that magnet-slot pair: the asymmetric overlap is greatest, reluctance changes fastest, and dW/dθ is at its maximum for that pair. For 12s/4p (LCM = 12), there are 12 such events per revolution; for 12s/10p (LCM = 60), there are 60.

These alignment events are distributed around the circumference at different rotor angles, so at any given instant most pairs are not at their alignment peak — they are somewhere in the middle of their transition cycle and contribute a smaller, partially cancelling force. The net cogging torque is the sum of all contributions at that rotor angle. The worst-case position is the one where the greatest number of pairs are simultaneously at their alignment peak, adding their forces in the same direction. That maximum number of simultaneously coincident alignments is exactly GCD(Q, P).

Since GCD = Q × P / LCM, a higher LCM means a smaller GCD: at no rotor position do more than GCD pairs reach their alignment peak simultaneously. The force from each individual alignment event is set by the machine geometry — magnet strength, slot opening, air gap — and is the same regardless of the slot-pole combination. What the slot-pole combination controls is how many of those alignment events ever coincide at the same rotor angle. Fewer simultaneous alignments means a lower net peak cogging torque.

Low pole count motors (P = 4 to 8)

Industrial servo motors, spindle motors, and general-purpose drives typically use 4 to 8 poles to achieve a good balance of electrical frequency, iron loss, and winding complexity. For these machines, increasing the slot count directly increases the LCM and reduces cogging torque.

The 36s/8p combination stands out: its GCD is only 4 (not 8 as you might expect from an 8-pole machine), because 36 and 8 share only 4 as their largest common factor. The result is LCM = 72 — the highest in this table. This is why precision servo motor designs often land on 36 slots even when using a lower pole count.

Notice also that 12s/4p and 12s/6p both produce LCM = 12, equal to the slot count. This is a general result: when Q is an integer multiple of P (integer-slot winding with q = 1 slot per pole per phase), LCM(Q, P) = Q. Adding more slots always helps in this regime.

High pole count motors (hub motors, P ≥ 14)

Direct-drive hub motors, axial-flux motors, and outer-rotor machines for robotics typically use 14 to 50 poles to generate high torque at low speed without a gearbox. These motors almost universally use fractional-slot concentrated windings (FSCW), where the number of slots per pole per phase q is less than 1. FSCW motors naturally produce very high LCM values because Q and P are close in magnitude and share few common factors.

The 54s/46p combination is remarkable: GCD(54, 46) = 2, so LCM = 54 × 46 / 2 = 1242. Over 1200 cogging cycles per revolution means each individual cycle is less than a third of a degree wide. A motor built to this combination has essentially imperceptible cogging at the shaft.

The intuition behind why FSCW high-pole motors do so well: when Q and P are close in value — differing by only 4 to 8 — and their GCD is small (ideally 2), the LCM approaches Q × P / 2. For a 36-slot, 40-pole machine: 36 × 40 / 4 = 360. The two numbers are nearly equal, so their ratio Q/P ≈ 1, and neither divides evenly into the other.

Secondary levers: slot opening and magnet embrace

Once the slot-pole combination is fixed, two geometric parameters offer further reduction in cogging torque. Neither can compensate for a poor LCM, but together they can reduce cogging amplitude by another 30–50% compared to a non-optimised geometry.

Slot opening width (b_s0). The slot opening is the gap at the top of the slot through which the winding is inserted. A wider opening creates a larger and more abrupt reluctance variation as the magnet edge passes over it, increasing cogging amplitude. Narrowing the slot opening smooths this transition. In practice, the opening cannot be reduced below about 1–1.5 mm for winding insertion, and very narrow openings increase slot leakage inductance. The slot opening should be no wider than necessary for manufacturing.

Magnet pole embrace (α_p). The magnet embrace is the ratio of the magnet arc length to the pole pitch. A full embrace (α_p = 1.0) means the magnet spans the entire pole, with magnet edges at the slot centrelines. This is not optimal for cogging: both leading and trailing edges of the magnet create cogging contributions that add in phase. At a specific embrace below 1.0 — typically in the range 0.70 to 0.80 — the two edge contributions arrive out of phase with each other and partially cancel. The optimal value depends on the slot-pole geometry and must be found by parametric simulation or optimisation, but starting from α_p ≈ 0.75 and sweeping in increments of 0.02 is a reliable approach.

Summary

Cogging torque arises because the rotor magnets are attracted to positions of minimum magnetic reluctance — directly over the stator teeth. The number of cogging cycles per revolution is LCM(Q, P): more cycles means lower peak amplitude. For low-pole motors, increasing the slot count raises the LCM. For high-pole hub motors, fractional-slot concentrated windings with Q and P close in value and sharing few common factors naturally achieve LCM values in the hundreds. Once the slot-pole combination is set, narrowing the slot opening and optimising the magnet embrace trim the remaining cogging further.