Cogging torque

where W_g is the magnet energy per pole stored in the air gap, equal to 0.5B_gH_gA_gg.

This energy is determined by the volume of the air gap and the flux density B_g, so in order to minimize the volume of magnet material required, it appears that the magnet should be operated with the maximum energy product B_MH_M . If the demagnetization characteristic is straight, then the maximum energy product occurs when B_M =0.5B_r, with a permeance coefficient approximately equals to 1, i.e., the operating point is halfway down the demagnetization characteristic. However, this theoretical result is seldom applied in practical machine design. The reasons are: the allowances needed for the demagnetizing MMF of the phase currents and temperature effects. Nevertheless, it is still meaningful to talk about a magnet material as having a high maximum energy product (BH)_max, because this is a single number representing the fact that both the remanent flux density and the coercivity are high. In common parlance, the (BH)_max figure is widely used to express the strength of various magnet grades.

the motor is running, additional oscillatory torque components can result from the

interaction of the magnet with space harmonics of the winding layout and with current harmonics in the drive current. These additional oscillatory torque components are electromagnetic and are generally referred to as torque ripple, while the term cogging is often reserved for the zero-current condition. In a well designed machine, the torque ripple and the cogging should both be negligible, but it is possible for the torque ripple to exceed the cogging torque by a large amount if the motor has an inappropriate combination of winding layout, drive current and internal geometry.

With a large number of slots per pole the cogging torque is inherently reduced by the fact that the relative reluctance variation seen by the magnets is reduced as it successively covers and uncovers the slots one at a time. In fact, the reluctance variation can be thought of as being concentrated at the edges of the magnet. A small amount of skew is then usually sufficient to eliminate most of the cogging. When the number of slots per pole is closer to 1, the slot geometry becomes more important and the widths of the teeth in particular can be adjusted to minimize the cogging effect.

2.4.1 Cogging torque calculation approaches

Cogging torque calculations techniques fall into two primary categories: analytical and numerical [25]. Analytical approaches typically begin with calculation of the air gap magnetic flux density distribution [26], [27], [28] and others. These calculations invariably require a series of simplifying approximations in order to make the problem more tractable, such as assumptions of infinite iron permeability and/or zero flux density under the stator slots. Cogging torque is then derived from the flux density distribution either by taking the derivative of the associated co-energy or by summing the lateral magnetic force along the sides of the stator teeth.

Finite element analysis is the most popular approach for numerical calculation of the cogging torque [29], [30], [31] and others. However, closer examination indicates that there are many variations in how finite element analysis is used. Although the majority of reported studies have used well-established 2-dimensional (2-D) finite element analysis,

more recent work has applied 3-dimensional (3-D) formulations in efforts to model the

influence of machine end effects and structure skewing more accurately. Torque is typically extracted from the finite element analysis using one of two approaches:

1) the energy method by which the torque is calculated from differences in the magnetic vector potential at incremental angular positions.

2) more direct torque calculation by integrating the resulting Maxwell stress tensor over the machine’s air gap surface.

Reported results suggest that no one of these cogging torque calculation methods can be judged superior to the others under all conditions. Although the finite element analysis techniques offer opportunities for more precise field calculations than the analytical approaches based on simplifying assumptions, the finite element results are also subjected to errors introduced by mesh generation problems or inadequate models of the iron and magnetic characteristics. Given the complicated machine geometries and associated nonlinearities in material properties, there is a general trade-off between calculation complexity and required torque prediction accuracy, which must be judged on a case-by-case basis.

2.4.2 Cogging torque minimization techniques

An extensive variety of techniques for minimizing cogging torque is documented in the literature for both sinusoidal and trapezoidal permanent magnet machines. The majority of this work has been carried out during the last decade coincident with the growing interest in brushless permanent magnet machines for high performance applications [25].

One of the most effective and familiar technique for cogging torque minimization is stator slot skewing. Several studies have demonstrated that skewing the stator slots by one stator tooth pitch can reduce the cogging torque to very low levels [32], [33] and [34]. If stator skewing poses unacceptable manufacturability problems, the alternative approach of skewing the rotor magnetic field distribution via either skewing rotor magnet magnetization or skewing the discrete magnet segments on the rotor.

Because the air gap magnetic permeance variations caused by stator slots are such an

important factor in cogging torque generation, a variety of additional techniques have been suggested for minimizing these variations or at least favorably modifying their harmonic spectra. One of the more obvious of these approaches is minimization of the stator slot openings [35] and [36]. Taking this approach a step further, slotless stator configurations have been adopted in applications where total elimination of cogging torque is required.

Other techniques seek to reduce cogging torque production by pushing the harmonic components in the spatial air gap permeance distribution to higher frequencies by either adding dummy slots or dummy teeth to the stator laminations. Alternatively, this frequency spectrum can be beneficially modified to reduce cogging torque either by shifting the angular positions of individual stator slots or by adopting a fractional number of slots per pole.

Alternative approaches for reducing cogging torque have been investigated with unskewed stators. For example, optimal ratios of magnet arc width to pole pitch have been identified, combined with a strategy of shifting alternate magnet arcs by one-half stator slot pitch in multi-pole designs [29], [33] and [37]. However, the effectiveness of these techniques is dependent on maintaining accurate mechanical tolerances on the physical dimensions and magnetization of the rotor magnets. Other rotor-based techniques, which have been proposed for reducing cogging torque, include shaping of the rotor magnet segments and addition of thin magnet retaining ring. In this thesis, it is shown that slot opening filling with bonded soft magnetic material of relative permeability higher than that of air is another possible approach to reduce the cogging torque.

Chapter 3