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Analytical modeling of the cogging torque in permanent magnet motors

Analytische Modellierung der Einrastmomente in permanentmagneterregten Motoren

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Contents

An exact analytical description of the magnetic field in idealized models of surface mounted permanent magnet motors is presented. An approximate closed form solution is derived therefrom. As to the cogging torque results obtained from measurements, finite element calculations, exact and approximate analytical modeling are compared with each other. The mutual dependency of armature geometry and magnetization pattern as to their influence on the cogging torque is discussed.

Übersicht

Es wird eine exakte analytische Beschreibung des Magnetfeldes für idealisierte Modelle von permanentmagneterregten Motoren mit am Lufspalt montierten Magneten angegeben. Daraus wird eine näherungsweise Lösung in geschlossener Form abgeleitet. Bezüglich der Einrastmomente werden Ergebnisse von Messungen, Finite-Elemente-Berechnungen, exakter und näherungsweiser analytischer Modellierung miteinander verglichen. Die wechselseitige Abhängigkeit von Ankergeometrie und Magnetisierungsmuster bezüglich ihres Einflusses auf die Einrastmomente wird diskutiert.

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Abbreviations

A :

vector potential

A :

z-component of the vector potential

A I,A II,A III :

z-component of the vector potential in slot numberl (region I), airgap (region II), and magnet (region III)

a I0 ,a II0 ,a III0 :

average value ofA I,A II,A III

c l m :

fourier series coefficients ofA I

a II k ,b II k ,c II k ,d II k :

fourier series coefficients ofA II

a III k ,c III k ,a III n ,c III n :

fourier seris coefficients ofA III

B :

flux density

B x ,B y :

flux density components inxy-coordinates

B r ,B ϕ :

flux density components in rϕ-coordinates

B I r ,B I ϕ :

flux density components in slot numberl (region I)

B II r ,B II ϕ :

flux density components in airgap (region II)

B III r ,B III ϕ :

flux density components in magnet (region III)

H :

magnetic field

H x ,H y :

magnetic field components inxy-coodinates

H r ,H ϕ :

magnetic field components inrϕ-coordinates

H I ϕ ,H II ϕ ,H III ϕ :

ϕ-component of the magnetic field in slot numberl (region I), airgap (region II), and magnet (region III)

M :

magnetization

M x ,M y :

magnetization components inxy-coordinates

M r ,M ϕ :

magnetization components inrϕ-coordinates

M k :

fourier series coefficients ofM r (magnet reference frame)

a M k ,c M k ,a M n ,c M n :

fourier series coefficients ofM r (armature reference frame)

e k ,e n :

auxiliary quantities describingM r

μ0 :

permeability of air

B R :

remanence of permanent magnet

ε:

width of transition zones between north and south poles divided by pole pitch (cf. Fig. 7)

P m :

number of magnet pole pairs

P :

fundamental order of cogging torque, equal to the smalles common multiple of 2P m andQ s

Q s :

number of slots

x, y :

cartesian coodinates, armature reference frame

r,ϕ:

polar coordinates, armature reference frame

θ:

angle of rotation of the magnet

e x,e y,e z :

unit vectors (cartesian coordinates)

e r,e ϕ,e z :

unit vectors (polar coordinates)

r 1,x 1 :

radius/position of slot bottom

r 2,x 2 :

radius/position of armature-airgap interface

r 3,x 3 :

radius/position of magnet-airgap interface

r 4,x 4 :

radius/position of magnet-yoke interface

ϕpyl :

lower border of slot numberl (xy/rϕ-coordinates)

w 1,W 1 :

width of slot numberl (xy/rϕ-coordinates)

w, W :

slot width for symmetric armature (xy/rϕ-coordinates)

α:

tooth width divided by tooth pitch (cf. Eq. (61))

f(k, n, ±):

auxiliary functions describing the amplitude reduction of fourier series coefficients of field quantities due to slotting (cf. Eqs. (58), (59), and (60))

f(k,n,±) :

auxiliary functions describing the coupling between fourier series coefficients of field quantities due to slotting (cf. Eqs. (50) and (51))

R, S k (a,b), C k (a,b), T k (a,b) :

auxiliary functions defined in Table 1

L y :

length of linear motor

L :

motor axial length

T c :

cogging torque

T v :

fourier series coefficients of cogging torque

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Authors and Affiliations

  1. Philips GmbH, Forschungslaboratorien Aachen, Postfach 1980, D-52021, Aachen, Germany

    B. Ackermann

  2. HEAD acoustics GmbH, Kaiserstraße 100, D-52134, Herzogenrath, Germany

    R. Sottek

Authors
  1. B. Ackermann
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  2. R. Sottek
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Ackermann, B., Sottek, R. Analytical modeling of the cogging torque in permanent magnet motors. Electrical Engineering 78, 117–125 (1995). https://doi.org/10.1007/BF01245643

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