Analytical Analysis of Cogging Torque in Permanent Magnet Machines With Unequal North and South Poles, With Particular Reference to Consequent Pole Machines
Ji Qi, Zi Qiang Zhu, Luocheng Yan, G.W. Jewell, Chengwei Gan, Yuan Ren, Simon Brockway, Chris Hilton
Abstract
In this paper, a general analytical model is developed to analyze the cogging torque accounting for unequal north and south poles as well as pole shifting, which can be applied to consequent pole permanent magnet (CPPM) machines and other rotor PM machines. The optimal slot opening and optimal pole-arc to pole-pitch ratio with/without pole shifting for minimum cogging torque are then analytically derived and verified by finite element analyses and experiments. It shows that in contrast to conventional PM machines with equal north and south poles, of which the fundamental cogging torque order is the least common multiple (LCM) between slot number <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N<sub>s</sub></i> and pole number 2 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> , the fundamental cogging torque order for the PM machines with unequal north and south poles (as often the case in CPPM machines), is the LCM between <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N<sub>s</sub></i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> , which has been derived analytically. However, in the special case when the slot number in one submachine is even, the LCM between <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N<sub>s</sub></i> and 2 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> has the same value as the LCM between <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N<sub>s</sub></i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> .