# Optimum Slot and Pole Design for Vibration Reduction in Permanent Magnet Synchronous Motors
## 1 Introduction  
Permanent Magnet Synchronous Motors are one of the most widely used electric motor types [1], with applications in a wide range of sectors including automotive, railway transportation, and elevators, as they provide great torque capacity. However, one disadvantage is that in those applications, comfort of users is a concern, and thus, the noise and vibration generated should be low. For this reason, it is necessary to optimise the design of the electric motor, taking into account the vibration response. The optimisation of the design of an electric machine is a very challenging task due to the consideration of multiphysics analysis [2]. In recent years, a number of research studies have been carried out to improve the design of PMSMs, analysing factors such as magnet eddy current loss, torque characteristics (average torque level, cogging torque, torque ripple, and total harmonic distortions of back EMF), radial forces, vibration, noise, and thermal issues. Several design improvements have been proposed regarding the torque ripple or cogging torque minimisation: the shape of the magnets was optimised in [3], notches were introduced on the permanent magnets in [4], the angle of the stator tooth was improved in [5], the shape of the tip of the tooth was enhanced in [6, 7], and auxiliary teeth were added in [8]. Pole and slot number combinations were also analysed in [9] to minimise the cogging torque and the Unbalanced Magnetic Force (UMF). In fact, a very low cogging torque can be obtained if the slot and pole numbers are chosen so that the Least Common Multiple (LCM) is large, as observed in [10]. The closer the number of slots (\(Qs\)) to thenumber of poles (\(p\)), the higher their LCM. However, a low cogging torque does not always guarantee a low torque ripple. Regarding the reduction of vibrations and noise, the magnetisation direction of the magnets and the length of the air-gap [11], the width of the slots and the width of the tooth tip [12; 13], and the shape of the magnets [14; 15] were analysed. In a further investigation, the shape of the tooth, including notches [11], was studied. In addition, the influence of the yoke thickness, the tooth shape, and the radius of the junction tooth/yoke were examined to determine which of those change the mode shapes and which mode shapes are excited by electromagnetic forces [16; 17]. The number of slots and the number of pole pairs are the key design parameters for the electromagnetic and vibratory performance of a PMSM [18]. They influence not only the frequencies of the excitation forces, but also which modes of vibration are excited. Comprehensive analyses of the forces together with structural and acoustic calculation procedures [19] showed that the interaction of permanent magnet field and stator slotting contributes the most to electromagnetic noise. Taking this into account, a noise reduction strategy was proposed to optimise the slot opening width. In the work of [20], the vibrations generated according to the spatial distributions of the forces were studied for several pole and slot combinations. However, concentrated forces were applied on the teeth, hence, the contributions of a significant number of harmonics were not taken into account due to spatial aliasing. More in depth analyses of pole and slot combinations were carried out in [13; 21; 22; 23] with significant conclusions. These authors observed that the contribution to noise of the pressure harmonic of spatial order 0 increases with the number of rotor poles. Configurations resulting in low noise were those in which low-order force harmonics were found to be weaker [21; 22]. In addition, configurations with a Greatest Common Divider (GCD) of the number of slots and poles equal to 1 were not recommended due to the unbalanced magnetic force [9; 10]. The dominant frequency harmonic is mainly generated by the interaction of the permanent magnet field and the armature reaction field.