Electromagnetic Model for Evaluation of Flux Harmonics and Resulting Magnetic Forces in Induction Motors

8/10/2019 Electromagnetic Model for Evaluation of Flux Harmonics and Resulting Magnetic Forces in Induction Motors

1/6


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541

general induction motor configuration

is shown in Fig.

1.

The

r-8

cross

sectional plan of t he whole motor

region is divided into subregions as

shown. To determine the harmonics

due to stator and rotor slots and the

rotor relative motion, t he following

assumptions a re made:

(1) The s tator winding is replaced by

an equivalent current sheet.

For

a

three-phase machine t he general

expression for the stator excitation

current density is:

JJ0.t) = J,(n

,m )

d @dmoot (1)

where

oo

is the supply frequency,

8

is the angle subtended at th e center

of the machine (in mechanical

degrees) and p is the number of pole

pairs.

JS(n,m)

is the amplitude of

the spatial harmonic of order n

which is determined by the winding

distributi on function, and the

temporal harmonic of order

m,

which

is determined by the nature of the

excitation current. In this work it

is assumed that the winding is

a

sinusoidally distributed balanced

three-phase winding,

so

that n and

m

take only the value 1 The amplitude

of the fundamental component of the

excitation current is determined from

the winding distribution function.

(2)

The stator slots are replaced by

a region of thickness equal to the

slot depth. This region has a

periodic permeability function

p(8)

.

(3) The rotor slots are replaced by a

layer on the top of the rotor back

iron region. This layer has a

thickness equal to the rotor slot

depth and a periodic conductivity

functio n o(8, t) that is also a

function of time since the rotor

r o t a t e s w i t h s p e e d

0

a n d

conductivity

O 8 ,

t)

The periodic permeability p ( 0 ) and

conductivity U(8,t) functions can be

expressed in terms of Fourier series:

m a

p@) = b(n)

ein se

n

O(8.t)

=

a n)

dnN& e i f lromt

(3)

n

Here, the number of stator slots,

Ns,

determines th e periodicity of

p(8):

the number of rotor slots, Nr,

determines the angular periodicity of

O(8,t) and the rotor mechanical

speed,

om,

determines its tempo ral

periodicity. The amplitudes of the

harmonic components b(n) and a(n) are

obtained for a rectangular shape

slots as:

a O) =

03

h

( 5 )

where W is the slot width, t is the

tooth width, px is the pitch and the

subscript x=s and x-r refers to

stator and rotor respectively.

ELECTR-TIC

FIELD

EQUATIONS

From Maxwell's equations, t he general

differential equation describing the

axial component of the magnetic

vector potential A in the different

regions of the motor is the Helmholtz

equation:

2

V A = p[Js + J e ]

where Js is the stator excitation

surface current density and Je is the

rotor induced current density, which

is related to

A

by:

J, = -Q - CO,,,-

[

(7)

From

Eqs. 6

and

I

it can be

conjectured that

A

will contain

harmonic components of the same

temporal and spatial frequencies as

Js(8,t),

p ( 0 )

and o(8,t),

so

that

a

may generally be written in the form:

m.1

8 )

The index i=1,2,3,4 is used to denote

the different regions within the


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542

motor. The indexes m and 1 represent

the effects of the harmonics of the

rotor and stator slotted structures,

respectively. They contribu te to an

electromagnetic field with spatial

frequency

( m

Nr+l Ns+p) or effective

number of poles, and temporal fre-

quency (mNr%+Oo). The amplitu de

Ai(m,l

)

is a function of the radial

distance

r

within region i. For the

Helmholtz equation t o be satisfied,

these amplitudes must have the

following dependence on r:

Ai(m,l

=

ci(m.1) P m J ) Di(m.1)

r-M(mJ

(9)

w h e r e

M(m,I)dmN,+l +pl

and the

constants Ci(m,l) and Di(m,l) are to be

determined from the boundary condi-

tions at the three surfaces

separati ng th e four regions. (Fig. 1)

as well as at r=O and

r=-.

The

continuity of the normal components

of the magnetic flux density

B r

and

the discontinuity of the tangential

components of the magnetic field

intensity He at the surfaces r= ra,

rsl and rs2 result in six equations

relating the values of Ai(i=1,2,3,4),

the parameters of the four regions

and the excitation and the induced

current sheets. Substituting for ai,

A2,

543, A4, and Je using Eqs. 7 - 8 ) ,

and equating each of the harmonics of

the same order on both sides of the

boundary conditions, six equations

relating the six unknown amplitudes

C1, C2, D2, C3, D3, D4 are obtained.

By a series of substitutions they can

be reduced to a single equation

relating the amplitudes of the

harmonic c omponents of C1

m , l ) :

J

The functions y1-y6 and F are defined

in terms of the machine main

dimensions, parameters and the stator

excitation current, see Appendix (I).

Fina lly t o sol ve Eq. 10 for Cl(m,l)

the summation terms are cast in the

matrix form

:

Y(n-a Cl(n-)=F(n )

(11)

n

where

Y ( n , n )

and F(n

)

are known

elements

of

matrices described in

Appendix (1) and Cl(n') are elements

of a column matrix that is simply a

stacking of the variables

I

[Cl

m , l ) ,

1

=

fl,f2,f3,

...

rDTf1,f2,f3,

.... .

This matrix equation can be solved

for th e unkn owns Cl(n'), from which

the ampl itud es Cl(m,l) of t he dif-

ferent harmonic components are

obta ined . Once Cl(m,l) are deter-

mined, the other amplit udes can be

computed by use of the boundary

condition relations and thus the

magnetic vector potential can be

determined in all regions of the

motor.

Flux

density diattibution

Knowing t he magnetic vector potential

distribution within motor different

regions, Lz1, L I Z , A3 and A4, the two

components of the magnetic flux

density

3

and

Be

are defined

everywhere in the defined space by:

(12)

Generally

&

nd can be written in

the form:

IB e,t) =

B,(m.l) d( r+ +P) d mA+cla)

m.1

(13)

a e , t ) = Be(m,l )

d( N,+ +P)

d(fiA+@t

(14)

m.1

For r-rsl (th e stator surf ace) the

amplitudes of the harmonics of the

radial and the tangential field

components are:

(15)


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543

B e ( d )=E

M[ 2 (m. l )

D2(m.i)

~ y l ]

m.1

(16)

Knowing the amplitudes

B ( m , l )

and

the space and t he time frequencies of

each harmonic component, complete

information about the air gap flux

harmonic contaminatio n is obtained.

Force

d i s t r i b u t i o n a l o n g s t a t o r

surface

Knowing th e two components of the

magnetic flux density

9

and

Be

at

the stator surface (r=rsl), the

radial and the tangential magnetic

force density distributions are

obt aine d usin g of Maxwell's stre ss

formula,

Fe

=-

(17)

cb

Substituting for & and B r using Eqs.

13,14, the Fourier series expression

for the tangential force Fe is :

Similarly the radial force density

is

:

B ~ ( m i ~ l i ) B ~ ( m i - m ~ l i - ~ - B ~ ( m i , l i ) B ~ ( m i - m , l i - l )

mi, l i

(19)

where

B

conjugates of & and Be.

and Bi are the complex

MODEL FOR CALCULATIONS

As an example of t he capability

of

this approach to provide a wide range

of information, the amplit udes and

the frequencies of the generated

harmonics can b e computed for both

the radial and the tangential flux

and force components. The flux and

the forces spectra as well as their

distributions can be plotted along

any radial surface within the

machine. The theory develop ed in

this work, has been applied to a

typical 3-phase, 4-pole cage motor

with a stator and rotor slot number

of

40

and 3 1 respectively. The spa-

tial frequency spectra of the radial

flux at th e rotor surface is shown in

Fig.

2,

while the temporal distri-

bution is shown in Fig. 3. Good

agreement of t he waveforms with those

of previous researchers is in

evidence [l-51. The spatial harmonic

spectrum is shown for the tangential

flux component at the rotor surface

in Fig.

4

and for the temporal

distribution in Fig. 5. n important

capability of this approach is the

ability of predict not only the

frequency of th e force components but

also their amplitudes. The temporal

distribution of the radial and the

tangential force components at the

stator surface are shown in Figs. 6

and

I

respectively.

CONCLUSIONS

An analytical technique has been

developed for determining the

frequencies and amplitudes of the

spatial and temporal harmonic

components of the electromagnetic

field in induction motors, and their

associated torsional and radial

forces. The relative strengths of

these harmonics are sensitive to the

geometry. The technique can be used

to facilitate the analysis of

induction motors.

REFERENCES

[l] Binns, K.J. and G. Rowlands-Rees,

Simple rules for the elimination of cogging

torque in squirrel cage induction motors .

Proc. IEE. Vol. 121, No.

1,

Jan. 1974.

[2] Binns,

K.J.

and E. Schmid, \ Some

concepts involved in the analysis of the

magnetic fields in cage induction machines .

Proc. IEE. Vol. 122, No. 2, Feb. 1975.

[3] Binns, K.J. and G. Rowlands-Rees,

Main flux pulsation h tangential tooth

ripple forces in induction mtors . IEE. Vol.

122 No. 3, March 1975.

[4]

Binns, K.J. and G. Rowlands-Rees,

Radial tooth ripple forces in induction

mtors due to the mi n flux . Proc. IEE. Vol.

125, No. 11, Nov. 1978.

[ ] Binns, K . J . and P.A. Kahan, Effect of

load on the tangential force pulsations and

harmonic torques of squirrel cage induction

motors , Proc. IEE, Vol. 128, Pt.

B ,

NO. I

July 1982.


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544

APPENDIX

I)

A s

a n e xa m pl e o f t h e

y

a n d t h e F

f u n c ti o n s i n

Eq 10

F(m.l)

= Js

1) Ars

6 (m )

M

where

M2= M(m-n

J)

= (m-n)N,+IN,+p I

M3=

M ( m J - n )

= I(m)N,+(l-n

) +PI

APPENDIX 11)

To w r i t e Eq.

1 0 i n t h e g e n e r a l m a t r ix

form

(Eq.

l l , t h e s i x t e r m s o f t h e

e q u a t i o n c a n be r e w r i t t e n a s f o ll o w s:

( m , l> y l ( m , l ) =

C(m-m'J-l') y l ( m . ~

m )

')

m , i

where the 6 f u n c t i o n i s d e f i n e d

as:

6(0)=1

Adding t h e t e r m s

1

t o 6 Eq. 1 0

is

c o n v er t e d t o t h e f or m:

C(m-m

J- l )

Y(mJm J ) =

F(m.0

* I

m I

R e l a t i n g t h e d o u b l e a r r a y a r gu m en ts

m- m , 1-1

t o a s i n g l e a r gu me nt n

and

m,l

t o

n ,

t he above equa t i on can

be r e w r i t t e n a s

Eq.

11.

t a t o r c o r e r e g io n

4

OK s l o t s r eg io n

i r g a p r e g i o n 2)

o r s l o t s

nduc t i ng l aye l

t o r r e g io n

1)

Fi g. 1 Motor main dimen sions


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545

0.06

Br

ra)

I

I

esla

0 .04

0.

Br r a

Tesli

0

0.

-0.

-0.

0.w50

o.01oo 0.0150

Time

(sec)

Fi g. 3 Temporal di st r i but i on of

t he radi al f l ux densi t y.

Fi g. 2 Spat i al harmoni c spect rumof

t he radi al f l ux densi t y.

0.2

Time (sec)

Fi g. 4 Spat i al harmoni c spect rumof Fi g. 5 Temporal di st r i but i on of

t he t angent i al f l ux densi t y. t he tangent i al f l ux densi t y.

80000

I

-20000 1 I

0.0050

0.0100 0.0150 0.0050 0.0100 0.0150

Time

(sec)

F b - 7Temporal di st r i but i on of t he

T i m e

(sec)

Fi g. 6 Temporal di st r i but i on of t he radi al

f orce densi t y at stator surf ace.

t angent i al f orce densi t y at

st at or surf ace.

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Electromagnetic Model for Evaluation of Flux Harmonics and Resulting Magnetic Forces in Induction Motors