
Brushless dc motor windings are similar to the windings in a multiplephase ac motor. But brushless motors also contain a permanent magnet rotor and a sensor to detect the rotor position to produce signals that control electronic switches for commutation. The most common position sensor is a Halleffect device. 

The shape of the magnetic flux density in the air gap of an assembled motor is an indicator of its cogging torque. An example are these three curves shown for one pole pair. Each theoretical curve represents different magnetization and polearc angles for the same air gap length, magnet thickness, and material. 

A typical nineslot, eightpole motor produces this cogging curve during one of 18 identical cycles in a single revolution (360 electrical degrees). Because it has four pole pairs, one mechanical revolution contains 72 cycles. Superimposing this curve on the shaft torque curve for one electrical revolution provides total motor ripple at a selected current level. For certain applications, current injected into the windings can cancel some harmonics to reduce cogging torque, and occasionally, it can become smaller than the bearing torque. 

A 12slot, 8pole motor generates three magneticflux distribution curves when using different magnetic materials, altering the thickness of the magnet, and adjusting the air gap so the peak flux density remains constant. 

Cogging torques are plotted from the three waveforms in Flux distribution: 12slot, eightpole motor. Small changes in the magnetic waveform can generate significant variations in cogging torque for each of the three waveforms, although all magnets produce nearly identical motor parameters and performance, otherwise. In some cases, the slot fill may be too high to handle larger diameter wires during attempts to minimize cogging, but usually, a combination of wire size and turns can be found without making the motor impossible to wind. 

The largest cogging torque waveform amplitude is the 12 slot in a family of curves for 6, 9, 12, and 15slot lamination motors. The curves indicate that for an eightpole motor with a 9slot lamination produces the least cogging torque. In practice, the 12slot lamination may be used when the means for developing a new lamination may not be readily available. When the application is sensitive to cogging torque, then skewing the magnets or the stator is an option but may result in a combination of more copper, iron, and magnetic material. 

The table Characterizing multiplepole motors shows that motors may be built with 4, 6, 8, or 12 rotor poles for an 18slot lamination, but these test results show that the motor with a 12pole rotor produces the highest amount of cogging torque for the same inner diameter and slot opening. 

Comparing the effects of even versus odd slots, cogging is highest for the 6pole rotor among 4, 6, 8, and 12pole configurations in a 15slot lamination. The slot opening and inner radius of the stator are again a constant and the same changes are made to the rotor as described for the 18slot lamination. The amplitudes for the 15slot compared to the 18slot are significantly different as are the air gap diameters, magnet material, motor geometry, and stack lengths. 

Cogging and torque are measured on a sixslot, eightpole motor with a ring magnet. Calculated data are plotted as the smooth curve and the measured data are plotted with circular markers. The data were obtained by running the motor with a servodrive at a constant speed of 0.5 rpm while the torque transducer provides an output voltage proportional to the cogging torque. In this case, the calculated values are conservative. 

Another configuration changes the slot opening for a given design and compares the cogging torque waveforms. This is shown for a 12slot, eightpole motor. The curve with the largest amplitude has a 4mm slot opening and the three other curves represent 3, 2, 1mm slot openings with decreasing amplitude. As the slot opening becomes smaller, the cogging torque also becomes smaller but at a faster rate. That is, the reduction is much greater going from a 2 to 1mm slot than going from a 4 to 3mm slot. For fractionalhorsepower motors, the slot opening is usually not smaller than 1.5 mm. In any case, there will always be some cogging, but it is up to the designer to decrease this as much as possible using these techniques. 
Cogging torque in dc brushless motors comes from variations in magnetic field density around a rotor's permanent magnets as they pass the nonuniform geometry of the slot openings in the stator. In applications such as servosystems and spindle drives, the pulsating speed that cogging generates can blemish machined surfaces or reduce position accuracy.
Unfortunately, classical electromagnetic calculations do not provide the data needed to determine how much cogging torque might develop in a new paper design. Although a complete finiteelement analysis may be an alternative to manual methods, it usually requires more project time than is available. In most cases, several prototypes must be made to measure and eventually reduce the cogging torque. Thus, it is critical to have a simple check list of major factors that determine cogging torque during the initial design procedure so several iterations can be made before finalizing the drawings.
Major factors affecting cogging torque include magnetic wave shapes, airgap length, slot opening, number of stator slots and rotor poles, skewing, copper fill, pole pitch, flux distribution or density, magnet volume, and material weight. Relationships between some of these factors, including electrical degrees/cycle, and cycles/rev for the most widely used motor configurations are conveniently shown in the table Characterizing multiplepole motors.
Analyzing the ripple torque for each type leads to a set of guidelines for new designs. For example, the table shows that the maximum number of cycles in one electrical cycle for a stator with an even number of slots can equal the number of slots itself. But for a motor with an odd number of slots, the number of cycles can be twice the number of slots.
Moreover, for a given frame size and type of lamination, slot and pole combinations as well as different pole arc to pitch ratios and magnetization, can produce different cogging torques. Keeping the number of poles on both rotor and stator ID and slot openings constant, and varying the number of slots, shows how cogging torque behaves for different slot and pole combinations.
The accompanying diagrams show how to combine these factors to arrive at a design that generates minimum cogging. In any case, skewing the magnets or the stator core often can lower cogging a bit more. When a design without skewing already shows minimal cogging, the skew angle required to reduce cogging below a particular value will be much smaller. Also, designing for a trapezoidal or nearly sinusoidal airgap waveform (made by varying the pole arc to polepitch ratio) is a common practice that often reduces cogging torque even further.
Portions of this article were contributed by Kartik Sitapati and Rob St. Germain, Kollmorgen Corp., Radford, VA 24141, (540) 6334124, Fax: (540) 7310847, www.kollmorgen.com