永磁同步电机故障诊断分析英文材料

3676IEEE TRANSACTIONS ON MAGNETICS,VOL.46,NO.9,SEPTEMBER 2010

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 ?ux density of salient pole synchronous permanent-magnet (PM)machines including the stator slotting effect.The approach to analyze the air-gap ?ux density based on the new model is to deter-mine the air-gap magnetomotive force (MMF)and the ?ux-path permeance function.We determine the ?ux-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 https://www.360docs.net/doc/fc8769402.html,ing the new model with stator slotting,we calculate the air-gap ?ux density distribution of different electric machines using ?nite-element methods to prove the accuracy of the new model.

Index Terms—Air-gap ?ux density,analytical solution,?nite-element method (FEM),?ux-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 ef?ciency.Magnetic ?eld 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 ?ux density distribution in the air-gap is es-sential for the accurate prediction of the motor performances.Magnet con?guration,magnetization direction,air-gap length,and the number of the pole and slot combination have signi?cant effects on the ?ux density distribution in the air-gap.Further,the presence of stator or rotor slots have a large in?uence on the air-gap magnetic ?eld distribution and therefore on the electro-magnetic torque.The consequences can be cogging torque and torque ripples causing vibration,noise,and speed ?uctuation.Therefore,the detailed knowledge of the ?eld distributions in the air-gap is of main importance for predicting and optimizing the performance of PM machines.

Of course as well known,slotting in?uences the magnetic ?eld in two ways.First,it reduces the total ?ux per pole,an effect which is usually accounted by introducing the Carter co-

ef?cient into the calculation.Second,it affects the distribu-tion of the ?ux 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 ?ux 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 ?gures in this paper are available online at https://www.360docs.net/doc/fc8769402.html,.

Digital Object Identi?er 10.1109/TMAG.2010.2049269

The air-gap magnetic ?eld with slotting effects can be eval-uated by a variety of techniques including analytical or semi-analytical methods as well as numerical techniques like ?nite 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 ?exible for the ?rst step of design stage of electrical machines.Analytical methods are useful tools for ?rst 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 signi?cant number of publications on the analytical solution of the air-gap magnetic ?eld 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 ?rst one concerns the use of con-formal mapping to consider slotting effects [1]–[7],and the ac-tual air-gap ?ux density is calculated by multiplying the rela-tive permeance function with the radial ?ux 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 ?ux density ?uctuation due to stator slotting,it doesn’t take correctly into consideration the slotting effect.The reason is that the magnetic ?eld 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 ?ux 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 ?ux path.However,even this modeling procedure describes accu-rately the air-gap permeance taking into account the stator and

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DAJAKU AND GERLING:TATOR SLOTTING EFFECT ON THE MAGNETIC FIELD DISTRIBUTION OF SALIENT POLE SYNCHRONOUS PM MACHINES

3677

Fig.1.(a)Simple PM machine with inset magnet in the rotor.(b)The corre-sponding?ux-path permeance function.

rotor slotting,it leads to an excessively complex expression for the total permeance function and therefore the obtained total air-gap permeance isn’t suitable for further mathematical op-eration during determination of the air-gap?ux density and also other parameters of electrical machines.

Therefore,in[18]a new and simple calculation model for the analysis of the salient pole PM machines is developed and pre-sented which is validly for different winding and rotor topolo-gies.The approach to analyze the air-gap?ux density based on the new model is to determine the air-gap magnetomotive force(MMF)and the?ux path permeance function.The air-gap MMF is described with the well-known Fourier series func-tion,however,the?ux-path permeance function is determined using a simple magnetic reluctance network.Determination of the air-gap?ux density with the new model considers the MMF harmonics and the rotor permeance harmonics.However,the stator slotting effect is neglected during this analysis.Therefore, the aim of this work was further to improve the new mathemat-ical model developed in[18]considering the stator slotting ef-fect during calculation of the air-gap?ux density.In analogy to [14]–[17]the stator slotting effect is considered during determi-nation of the total permeance function,however using a simple approximation.Further,the new model is valid for smoothed and saliency rotor topologies and also is useful for determining the air-gap?ux density due to stator and rotor excitation.The new analytical model with the stator slotting shows that for the electric machine with saliency rotor topology,the stator slotting effect over the rotor teeth and rotor slot region is different. II.A IR-G AP F LUX D ENSITY OF PM M ACHINES U SING THE

N EW C ALCULATION M ODEL

A simple two-pole PM machine with inset magnets in the rotor core is illustrated in Fig.1(a).However,Fig.1(b)shows the variation of the magnetic?ux-path permeance along the cir-cumferential direction for the given machine.During the?rst approximation of the permeance function,the stator slotting ef-fect is neglected.

According to[18],the approach to determine the air-gap?ux density based on the new model is to determine the air-gap mag-netomotive force(MMF)and the?ux path permeance function

as

(1)

Fig.2.Winding MMF distribution.(a)Distributed winding with q=2.

(b)Concentrated winding with q=0:4.

where is the?ux-path permeance

function,is

the air-gap MMF,

and is the angular displacement along the

stator circumference.The factor“2”in(1)is used to take into

consideration the magnetic coupling between the?ux density of

neighbor poles.

A.Air-Gap Magnetomotive Force(MMF)

For the PM machine,the air-gap MMF consists of two

components:stator MMF(current excitation)and rotor MMF

(magnet?ux).

Stator MMF.When the winding distribution is known,the

MMF created by the current can be described using the Fourier

series function.The MMF characteristics for different winding

types are widely studied in many literatures,e.g.,[20]to[22].A

general expression for the MMF distribution is presented in the

following(2).However,Fig.2show the MMF distribution and

corresponding space harmonics for two different winding

types.

(2)

where is number of

phases is the number of turns

per phase,is stator current(peak

value),is the load

angle,

is the angular

frequency,is the MMF space harmonic,

and

is the winding factor.

Rotor MMF;By the PM machines there are a lot of rotor

topologies that can be used and therefore the rotor MMF dis-

tribution depends on the rotor topology and also on the magne-

tization direction of the magnets.For a simple PM machine with

inset magnets in the rotor as is shown in the Fig.3(a)the rotor

MMF distribution is illustrated in Fig.3(b).A general expres-

sion for the rotor MMF distribution is presented in the following

(3):

(3)

where

and are space harmonics and corresponding am-

plitudes for the rotor MMF.