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system design and analysis. It shows that, in comparison to the fi nite element
method, the calculation of reluctance torque using the magnetic circuit method
produces fairly accurate result for a wide range of rotor position. However, for
a narrow range of rotor position near θ = 0, the magnetic circuit method gives
erroneous result. However, the magnetic circuit method gives an analytical
expression of the reluctance torque, which is very helpful in the analysis of the
torque performance of a device, and can simplify the computation.
The virtual work principle is used in the analysis described above. If the
magnetic fi elddistributionisknown,thismethodisaneffective tool for calcu-
lation of the electromagnetic torque/force, which include both the Ampere's
and reluctance torque/force.
4.2.4 Cogging Torque and Unbalanced Magnetic Pull
Putting slots on the stator core makes the installation of winding easy and
reliable. However, the slots also induce the cogging torque problem in the
PM motors. This torque is related to the shape of the slots, number of slots,
number of magnetic pole-pairs, and the magnetization of the magnets [109],
Figure 4.31: Cogging produced in a PM motor with 12-slots and 2-pole-pairs
Generation of cogging torque is explained here using the PM motor shown
in Figure 4.31. In this motor, the magnets are mounted on the inner surface of
the rotor. When the rotor rotates from position a to position b, the total teeth
surface facing the magnet is increased. Following the discussions in section on generation of reluctance torque, the increased surface area of the
teeth facing magnet means decrease in reluctance of the main magnetic circuit
when the rotor is rotated from position a to position b, and reluctance torque
is induced. It can be shown that, if the ratio between the slot number and
the magnetic pole is not an integer, the variation in reluctance is not very
signi fi cant in the rotation of motor, although it still exists to some extent [77].
In hybrid stepping motors [108], cogging torque can be used to produce holding
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