# Investigation on a choice of stator slot skew angle in brushless PM machines

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## Abstract

The paper investigates the effects of stator slot skewing in a permanent magnet brushless DC motor. A simple analytic formula for calculation of the best angle of stack skew, which leads to nearly total reduction of the cogging torque, is developed. The skew angle obtained from this formula is different to that used by the designers of PM brushless motors. The analysis is carried out for a fractional horsepower brushless permanent magnet motor with the surface-mounted magnets using a time-stepping, multi mesh-slice finite element model, to assess the impact of this change. The steady-state characteristics and core losses are analyzed quantitatively using the elaborated numerical model. It is shown that smaller skew angles obtained from the formula lead to noticeable rise in motor overall efficiency and decrease of the core loss. The possibility of accomplishment of the desired effect of skew in a real machine is also a subject of discussion.

### Keywords

Permanent magnet brushless DC motors Slot skew Core loss Electromagnetic torque Finite element analysis### List of symbols

- A
\(z\)th component of magnetic vector potential

- \(d\)
Thickness of electrical sheet

- \(f\)
Frequency

- \(H_{cu}\)
Components of coercive magnetic field vector, \(u=x,y\)

- \(i_u\)
Phase current, \(u=a,b,c\)

- \({{\varvec{v}}}_{\mathrm{s}}\)
Vector of branch voltage sources

- \(\ell \)
Length

- \(N\)
Finite element shape function

- \(n_{\mathrm{s}}\)
Number of slices (submodels) of multi mesh-slice model

- \(n\)
Time-step index

- \(n_{\mathrm{t}}\)
Number of turns

- \(L_{eu}\)
Inductance of phase end-winding, \(u=a,b,c\)

**L**Matrix of branch inductances

- \(p\)
Number of pole-pairs

- \(q\)
Ratio of pole-width to pole-pitch

- \(Q\)
Least number of slots in a doubly slotted machine

- \(Q_{\mathrm{s}}\)
Number of coils per phase of distributed winding

- \(R_u\)
Resistance of phase-belt, \(u=a,b,c\)

**R**Matrix of branch resistances

- \(S_n\)
Cross-sectional area of \(n\)th coil

- \(T\)
Time-period

- \(v_u\)
Supply voltage of \(u\)th phase-belt, \(u=a,b,c\)

- \(\alpha \)
Mechanical angle

- \(\alpha _{sk}\)
Skew angle

- \(\Delta t\)
Time-step length

- \(\lambda _u\)
Phase-belt flux linkage, \(u=a,b,c\)

- \(v\)
Magnetic reluctivity

- \(\varphi \)
Nodal value of magnetic vector potential

- \(\tau \)
Angular width of one slot-pitch

- \(\sigma \)
Electric conductivity

- \(\rho \)
Mass density

- \(\eta _n\)
Auxiliary coefficient equals 1 if current sense in \(n\)th coil is the same with \(z\) direction and \(-\)1, otherwise

- \(\eta _u\)
Auxiliary coefficient equals 1 if coil belongs to \(u\)th phase-belt and 0, otherwise

- \(\varOmega \)
Region of analysis

- \(\nabla \)
\(\left( {\partial /{\partial x},\partial /{\partial y},\partial /{\partial z}} \right)\).

## 1 Introduction

There are vast research works dealing with the cogging torque reduction issues in the permanent magnet machines, e.g. [1, 2, 3, 4, 5, 6, 7]. Although much has been done in this area to date, newer publications show that it is still of significant concern to the designers of permanent magnet machines [7, 8, 9, 10, 11]. Among many approaches developed to cope with this problem, the skew of the stator stack sheet pack is the most natural and perhaps, the simplest method of preventing the machine from generating the reluctance-type torques to apply at the design stage [1, 3, 7, 8, 9, 10, 11]. In permanent magnet machines, the major task of skew is to get rid of the cogging torque. In all machines, the angle of skew should be selected among values of mechanical angles between 0 and \(2\pi /Q\), where \(Q\) stands for the least number of slots on either stator or rotor side. A basic role of skew should be the cancellation of the most influential slot harmonic of the magnetic field which interacts with the fundamental stator mmf harmonic to produce the reluctance-type torque. An appropriate choice for the angle of skew can be made only after detailed investigation on the magnetic field distribution in the machine air-gap, because a principle of generation of the electromagnetic torque is complex [7].

In the above formula, the choice of most influential harmonic number is suggested by the mathematical formula without physical considerations on the magnetic field distribution in the motor air-gap. It can be shown that the best angle of skew can be even smaller than that suggested by (3) when the reference quantity of the considerations is the electromagnetic torque not the magnetic flux density. The skew angle given by (3) did not result in total reduction of the cogging torque. A much better value of the skew angle can be deduced in the following way.

Assume a permanent magnet machine at standstill with the number of poles \(2p\) and the number of slots \(Q\). For simplicity, also assume \(Q/p\) being integer. The order of the fundamental harmonic of the cogging torque in all permanent magnet machines (excluding the fractional-slot machines) is the least common multiple of 2 and number of slots divided by number of pole-pairs. In such a case, an obvious choice for skew angle would be angular width of one slot pitch, as it is done in (3). On the other hand, it is well known that the principle of generation of the electromagnetic torque is based on a cross product between all terms of Fourier series expansion of the distributions of radial (\(B_{\mathrm{r}}\)) and tangential (\(B_\psi \)) components of magnetic flux density. This is because an instantaneous value of the electromagnetic torque is in proportion to the product \(B_{\mathrm{r}} \cdot B_\psi \) [7]. This analytical operation results in an infinite sum of products of trigonometric functions. Each such product can be further expanded into a sum of two trigonometric functions of the sum and the difference between their arguments.

Relative skew angles calculated from (4) referred to slot-pitch width (integer-slot machines)

\(p\) | \(Q\) | |||||||
---|---|---|---|---|---|---|---|---|

12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | |

1 | 0.923 | 0.947 | 0.960 | 0.967 | 0.973 | 0.976 | 0.979 | 0.981 |

2 | 0.857 | \(\times \) | 0.923 | \(\times \) | 0.947 | \(\times \) | 0.960 | \(\times \) |

3 | \(\times \) | 0.857 | \(\times \) | \(\times \) | 0.923 | \(\times \) | \(\times \) | \(\times \) |

4 | \(\times \) | \(\times \) | 0.857 | \(\times \) | \(\times \) | \(\times \) | 0.923 | \(\times \) |

5 | \(\times \) | \(\times \) | \(\times \) | 0.857 | \(\times \) | \(\times \) | \(\times \) | \(\times \) |

6 | \(\times \) | 0.428 | \(\times \) | \(\times \) | 0.857 | \(\times \) | \(\times \) | \(\times \) |

7 | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) | 0.857 | \(\times \) | \(\times \) |

8 | \(\times \) | \(\times \) | 0.428 | \(\times \) | \(\times \) | \(\times \) | 0.857 | \(\times \) |

9 | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) | 0.857 |

10 | \(\times \) | \(\times \) | \(\times \) | 0.428 | \(\times \) | \(\times \) | \(\times \) | \(\times \) |

From the results summarized in Table 1, it can be observed that with increasing number of slots the angle of skew approaches the angular width of one slot-pitch, which is the most commonly used by designers. Skewing as a method of cogging torque reduction is usually applied to machines with distributed windings where the unity coil-span and distribution factor can be achieved relatively easily rather than in those with concentrated windings. This is due to the fundamental coil-span factor being less than unity and to the value of fundamental skew factor which has to be significantly smaller.

The skew affects all harmonics of the magnetic flux density, not only the slot harmonics, and the adequate analysis method must be involved to assess the results after application of skew angle resulted from (4). The skew factors resulted from this formula are slightly higher than those resulted from (3), and consequently it is worth investigating how application of new skew angle values influences the overall performance of machine as it may lead to cost-effective solutions. To characterize these effects quantitatively, the authors develop a quasi-3D multi mesh-slice finite element model [12]. An investigation is presented including the model development and the analysis of machine operation under no-load and full-load conditions. Additionally, the analysis of core loss is presented. The results are partially backed by measurements carried out on the laboratory test stand for a prototype outer rotor brushless DC motor, designed for a gearless drive of an electrically powered wheelchair.

## 2 Laboratory machine

Specifications of the prototype motor

Outer/inner rotor diameter | 194/ 170 mm/mm |

Outer/inner stator diameter | 168/110 mm/mm |

Length | 40 mm |

Rotor configuration | Outer with surface-mounted magnets |

Number of pole-pairs | 7 |

Number of stator slots | 42 |

Angle of stator skew | One slot-pitch |

Winding | Double layer, full pitch, 18 turns per slot |

Nominal voltage, power | 24 V, 200 W |

Nominal speed | 170 rpm |

## 3 Motor model for performance analysis

The transistor switches *S*1–*S*6 are modeled as two-state resistance elements, whilst the diodes as nonlinear voltage-dependent current sources considering rudimentary static characteristics. In this way, the model allows to estimate overall efficiency of the drive including inverter power loss.

The electromagnetic torque of a motor is calculated from the Maxwell stress tensor. The modeling of rotor movement by means of elements distortion and permutation of node numbers in the machine air-gap allows for computation of time variations of phase currents, voltages and torque. Such a model provides a transient solution. If he constant rotor speed is assumed, the steady-state solution is obtained after a few periods of phase current from the beginning of computations. From the steady-state waveforms the rms and average values can be computed to provide information on power balance and losses.

## 4 Analysis and results

### 4.1 Validation of the model

In the analysis the three lot-fill factors are considered. A first case (Fig. 6a, b) refers to the machine with specifications given in Table 2, with the slot fill factor equal to 0.778. The two remaining cases are referred to the machines having slot-fill factors equal to 0.6 and 0.9 (see Fig. 6c, d and Fig 6e, f, respectively). Owing to the fact that the torque is in proportion to a product \(B_\mathrm{r} B_\psi \), it can be seen in all figures that the most influential is harmonic \(Q/p+1\) (equal here to 7) with no matter what the slot-fill factor is. This confirms that Eq. (4) is the best choice to calculate the skew angle for such a motor.

### 4.2 Operation at no-load conditions

The skew by one slot pitch gives the value of fundamental skew factor of 0.955, whilst the second option gives \(k_{sk1}\) equal to 0.967. In the considered motor, the fundamental winding factor is equal to the skew factor due to unity fundamental coil-span and distribution factors.

A side effect of skewing is a reduction of the magnitude of axial component of current density by the factor of sec \(\alpha _{sk}\) which can be considered favorable in applications, where the demagnetization effects are particularly important.

## 5 Conclusions

Theoretical considerations and computations using a quasi-3D multi mesh-slice finite element model show that the best angle of skew of the sheet pack for the three-phase integer-slot permanent magnet machine with the surface-mounted magnets can be determined from a simple analytic formula. Application of a skew angle determined in such a way a very high reduction of electromagnetic torque pulsations caused by the angular variation of the equivalent magnetic reluctance of an air-gap.

The results of analysis have led to a few useful conclusions, but it is worth noting that these are based merely on a numerical experiment. The possibility of accomplishment of such an effect in a real machine is the matter of accuracy in the manufacturing process. This is because sensitivity of the magnitude of the cogging torque to the variation of the skew angle is very high.

The results presented were obtained for a fractional power machine due to limitations of laboratory facilities but these should be addressed particularly to larger machines where the problem of reduction of core loss and mechanical vibrations is crucial.

## Acknowledgments

This work was supported by the National Science Foundation under the Grant ECCS-0801671.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.