Stator Slotting Effect on the Magnetic Field Distribution of Salient Pole
Synchronous Permanent-Magnet Machines
Gurakuq Dajaku 1and Dieter Gerling 2
FEAAM GmbH,D-85577Neubiberg,Germany
Institute for Electrical Drives,University of Federal Defense Munich,D-85577Neubiberg,Germany
This paper deals with a new analytical method for determining the air-gap flux density of salient pole synchronous permanent-magnet (PM)machines including the stator slotting effect.The approach to analyze the air-gap flux density based on the new model is to deter-mine the air-gap magnetomotive force (MMF)and the flux-path permeance function.We determine the flux-path permeance function,which includes the stator slotting effect,by using a simple magnetic reluctance network that takes into account the geometry of the machine and the material ing the new model with stator slotting,we calculate the air-gap flux density distribution of different electric machines using finite-element methods to prove the accuracy of the new model.
Index Terms—Air-gap flux density,analytical solution,finite-element method (FEM),flux-path permeance,permanent-magnet (PM)machine,stator slotting effect.
I.I NTRODUCTION
P
ERMANENT-MAGNET synchronous machines (PMSMs)gain more and more importance for spe-cial drive applications.Up to recent years,PMSMs were known for small drives,e.g.,for servo applications.In the last years,PMSMs have been increasingly applied in several areas such as traction,automobiles,etc.This type of electrical machines offer many advantages,including a high power-to-weight ratio,constant power over a wide speed range,and high efficiency.Magnetic field analysis in PM machines is an important pre-requisite for the prediction of machine parameters,electromag-netic torque,radial forces,and so on.In particular,accurate knowledge of the flux density distribution in the air-gap is es-sential for the accurate prediction of the motor performances.Magnet configuration,magnetization direction,air-gap length,and the number of the pole and slot combination have significant effects on the flux density distribution in the air-gap.Further,the presence of stator or rotor slots have a large influence on the air-gap magnetic field distribution and therefore on the electro-magnetic torque.The consequences can be cogging torque and torque ripples causing vibration,noise,and speed fluctuation.Therefore,the detailed knowledge of the field distributions in the air-gap is of main importance for predicting and optimizing the performance of PM machines.
Of course as well known,slotting influences the magnetic field in two ways.First,it reduces the total flux per pole,an effect which is usually accounted by introducing the Carter co-
efficient into the calculation.Second,it affects the distribu-tion of the flux density in the air-gap of the electric machines.Therefore,the slotting effect is important to be taken into ac-count during determination of the air-gap flux density.
Manuscript received February 26,2010;revised April 09,2010;accepted April 15,2010.Date of publication May 03,2010;date of current version Au-gust 20,2010.Corresponding author:G.Dajaku (e-mail:Gurakuq.Dajaku@unibw.de).
Color versions of one or more of the figures in this paper are available online at .
Digital Object Identifier 10.1109/TMAG.2010.2049269
The air-gap magnetic field with slotting effects can be eval-uated by a variety of techniques including analytical or semi-analytical methods as well as numerical techniques like finite elements or boundary integral methods.Finite elements give accurate results considering geometric details and nonlinearity of magnetic materials.However,this method is computer time consuming and poorly flexible for the first step of design stage of electrical machines.Analytical methods are useful tools for first evaluation of electrical motors performances and for de-sign optimization since continuous derivatives issued from the analytical solution are of great importance in most optimization methods.A significant number of publications on the analytical solution of the air-gap magnetic field in slotted machines can be found in the literature,e.g.,[1]–[17].
Two analytical methods are mainly developed for modeling the stator slotting effect.The first one concerns the use of con-formal mapping to consider slotting effects [1]–[7],and the ac-tual air-gap flux density is calculated by multiplying the rela-tive permeance function with the radial flux density of a slotless motor.This method has an important drawback since it is as-sumed that the slot width is much smaller than its height.Fur-thermore,even though this method gives information about the air-gap flux density fluctuation due to stator slotting,it doesn’t take correctly into consideration the slotting effect.The reason is that the magnetic field and the slotting effect are determined separately and for different conditions.The second method con-sists of analytical solution of Laplace or Poisson equations in the different sub-domains (magnet,air-gap and slots)by ap-plying the boundary conditions on the interface between sub-do-mains [8]–[13].However,these models are not able to pro-vide an accurate evaluation of air-gap flux density if the both stator and rotor slotting have to be considered.Also as will be shown later,the rotor saliency effect on the stator slotting isn’t considered in these works.On the other hand,different from [1]–[13],in [14]–[17]another modeling way for the stator slot-ting is used;the slotting effect is taken into consideration during derivation of the total permeance function for the magnetic flux path.However,even this modeling procedure describes accu-rately the air-gap permeance taking into account the stator and
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