modelo dianamico de induccion lineal

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ISSN 1392 – 124X INFORMATION TECHNOLOGY AND CONTROL, 2005, Vol.34, No.1

DYNAMIC MODELS OF LINEAR INDUCTION DRIVE

R. Rinkevičienė, A. Petrovas

Vilnius Gediminas Technical University

Abstract. Due to open magnetic circuit and limited length of the linear induction motor inductor phase currents arenot equal. The article deals with model of linear induction drive taking into account magnetic asymmetry of linear

induction motor at different connection of windings. Developed models are presented and results of simulation areconsidered.

1. Indroduction 2. Mathematical model of a motor with Y

connected windings with neutral wireLinear electric motor (LIM) technology has been

broadly used in the aspects of traffic, industry, infor-mation, automation, material transportation, elevatorsand postal mechanical systems [1, 2, 3].

Three-phase motor with Y connected windings is presented in Figure 1. The figure shows voltages andcurrents in a circuit. The rotor of the motor is shown

turned by angle ϕ. Nevertheless by way of investigation broadeningand shrinking areas of application can be defined.

Due to worse efficiency linear electric drivesusually are designed for short time duty. Often a linear drive operates just in a dynamic mode and its para-

meters do not reach steady state values. Electromag-netic transients in the linear motor must be taken intoaccount in order to investigate characteristics of suchdrives. Due to open magnetic circuit of LIM asymmet-ry of currents takes place. The paper presents a modelof linear induction motor with consideration of elect-

ric and magnetic asymmetry due to open magnetic cir-cuit of LIM at different connection of windings. In amodel transform of coordinates is not used.

Figure 1. Y connected three-phase motor with neutral wire

According to traditions rotary or linear motors aremodelled by transformation three-phase motor toequivalent two-phase one and deriving equations in

the stationary reference frame or revolving (trans-

lating) reference frame. Indices α, β denote stationaryreference frame and u, v - reference frame translatingwith speed vk . If vk is equal to speed of magnetic fieldv0el , such reference frame is called x, y. Models of linear drives in transformed coordinate systems are

analysed in [4, 5, 6]. Modelling of control modes of drives with LIM meets difficulties to transform supply

voltage shape. The developed model has norestrictions on supply voltage shape and can be usedto model advance control modes. They do not requiretransform back model variables to real quantities.Models of linear motors are derived on an analogy between rotating and translating motors.

The system of equations describing rotating motor and LIM is formed from equations of voltage equilib-rium for inductor and secondary, flux linkages andequations to calculate electromagnetic torque and

force as well as equations to describe rotational (trans-lational) movement. Equations of voltage equilibriumfor one phase of the inductor winding and one phaseof secondary winding are:

+Ψ

=

+Ψ

=

,aaa

a

A A A

A

i Rdt

d u

i Rdt

d u

(1)

where u A , ua , – instantaneous values of inductor phaseA and phase a of secondary element voltages;

Ψ A ,Ψ a – flux linkages of inductor phase A and phase a of secondary element;

R A (L A ), Ra (La ) – A phase resistances (inductances)of inductor and secondary element; i A , ia , – instanta-

37




neous values of inductor phase A winding and phase asecondary element currents.

Phase A flux linkage of a stator and rotor is:

( )( ) ( )[

( )( ) ([

°++°−+++++=Ψ

°−+°++

++++=Ψ

,120cos120coscos

120cos120cos

cos

ϕ ϕ ϕ

ϕ ϕ

ϕ

C B ASR

cb Raaa

cbSR

aSRC BS A A A

iiiM

iiM i L

iiM

iM iiM i L

]

)]

))]

(2)

where M SR – mutual inductance between inductor and

rotor (secondary element) when the axes of coilscoincide [H]; M S ,(M R ) – mutual inductance between

different coils of inductor (rotor or secondary) in H; ϕ - angle between rotor and inductor coils in electricdegrees.

Electromagnetic torque developed by rotatingmotor is [6, 7]:

( )[ ]

( ) ([ ]( ) ([ .120sin

120sin

sin

°−++−

−°+++−

−++−=

ϕ

ϕ

ϕ

bC a Bc ASR

aC c Bb ASR

cC b Ba ASRem

iiiiii pM

iiiiii pM

iiiiii pM M

(3)

On a base of power equivalency in rotating andtranslating motors an expression to calculate the forcedeveloped by LIM is obtained:

, p

M F em

em ⋅=τ

π (4)

, x⋅=τ

π ϕ (5)

whereτ is pole pitch; p is the number of pole pairs, x-

excursion of the secondary.Figure 2 shows a scheme of linear induction motor,

where the angle of rotation ϕ is transformed to excur-

sion of secondary x .

Figure 2. Electrical circuit of linear induction motor windings

The dynamic model is supplemented by a move-ment equation.

3. Mathematical model of a motor with Y

connected windings without neutral wire

In motors with symmetrical windings suppliedwith balanced voltage system current does not flow inneutral wire. They can be modelled by the method

described above even if they are connected withoutneutral wire.

If phase windings of a motor are non-symmetricaland connected without neutral wire or if they are sym-

metrical but supplied by unbalanced voltage systemdue to absence of neutral wire they must be modelled

considering supply by line voltages.A scheme of a three-phase motor with Y connected

windings without neutral wire is given in Figure 3.

Figure 3. Scheme of a three-phase LIM with Y connectedwindings without neutral wire

Equations for an inductor and secondary are:

Ψ−−Ψ+=

Ψ−−

Ψ+=

Ψ−−

Ψ+=

Ψ−−

Ψ+=

,dt

d i Rdt

d i RU

dt

d i R

dt

d i RU

dt

d i R

dt

d i RU

dt

d i R

dt

d i RU

ccc

bbbbc

ccc

aaaac

C C C

B B B BC

C C C

A A A AC

, (6)

where , ,U , are the line voltages of

stator and rotor.

AC U BC U ac bcU

There are two independent currents in Y connec-tion. By Ohm’s law the third one is equal to the sum

of these two currents taken with minus sign. There-fore, one current of equation (6) can be expressed as asum others two. This statement is valid for flux lin-kages also. If two independent quantities are chosen,for example, currents of a and b windings, then currentand flux linkages of phase c are calculated as:

( ) ( )( ) ( )

Ψ+Ψ−=Ψ+−=

Ψ+Ψ−=Ψ+−=.,

,

bacbac

B AC B AC

iii

iii(7)

Substituting (3) to (2) yields:

( )( )

( )( )

( )( )

( )( )

Ψ+Ψ+++=

Ψ+Ψ+++=

Ψ+Ψ+++=

Ψ+Ψ+++=

.2

2

2

2

dt

d i R Ri RU

dt

d i Ri R RU

dt

d i R Ri RU

dt

d i Ri R RU

babcbacbc

babcacaac

B A BC B AC BC

B A B B AC A AC

(8)

38



Dynamic Models of Linear Induction Drives

Flux linkages of windings are calculated accordingto formulas, presented in [6]. Substituting these for-mulas to (8) and considering (7) yields:

;; 34 Ra Rb M L DM LC −=−=

xM DC B A SRτ

π cos32143 ==== ;

( )

( ) ( )

;3

sinsin3

3coscos3

2

dt

dx x

bi x

aiM

dt

di x

dt

di xM

dt

diM L

dt

diM L

i Ri R Ru

SR

baSR

BS B

AS A

B B AC A AC

++

−

−

++

+

+−+−+

+++=

π

τ

π

τ

π

τ

π

π

τ

π

τ

π (9)

;3

cos341

+==π

τ

π xM A D SR

;3

cos332

−== π τ π xM BC SR

;3

sinsin35

++

−=π

τ

π

τ

π xi xiM A baSR

;sin3

sin35

+

−−= xi xiM B baSRτ

π π

τ

π

( )

( ) ( )

;sin3

sin3

cos3

cos3

2

dt

dx xi xiM

dt

di x

dt

di xM

dt

diM L

dt

diM L

i R Ri Ru

baSR

baSR

BS B

AS A

BC B AC BC

+

−−

−

+

−+

+−+−+

+++=

τ

π π

τ

π

τ

π

τ

π π

τ

π (10)

;3

sinsin35

−+−=π

τ

π

τ

π xi xiM C B ASR

;sin3

sin35

+

+−= xi xiM D B ASRτ

π π

τ

π

The left-hand side of the equation is expressed as:

( )

( )( )

( )( )

−

+−−

−+−

+−−

−+−

=

st em

bcbaccb

bccaac

BC B AC BC

BC AC A AC

F F

v

i R Ri RU

i R R RU

i R Ri RU

i Ri R RU

xt ,F . (13)

( )

( ) ( )

;3

sinsin3

2

3cos3

cos3

dt

dx xi xiM

dt

diM L

dt

diM L

dt

di xM

dt

di xM i Ri R RU

B ASR

b Rb

a Ra

BSR

ASRbcacaac

−+−

−−+−+

+

−+

++++=

π

τ

π

τ

π

τ

π

π

τ

π

τ

π

(11)

Electromagnetic force is calculated from the expression:

( )

( )

( ) .120 x siniiiiiiM p

120 x siniiiiii pM

x siniiiiii pM F

bC a Bc ASR

aC c Bb ASR

cC b Ba ASRem

°−++−

−

°+++−

−

++−=

τ

π

τ

π

τ π

τ π

(14)

( )

( ) ( )

.sin3

sin3

2

cos3

3cos3

dt

dx xi xiM

dt

diM L

dt

diM L

dt

di xM

dt di xM i R Ri RU

B ASR

b Rb

a Ra

BSR

ASRbcbacbc

+

+−

−−+−+

++

+

++++=

τ

π π

τ

π

τ

π

τ

π

π τ π

(12)

4. Mathematical model of an induction motor

with ∆ connected windingsThe system of equations (9) – (12) describes in-

duction motor with Y connected winding withoutneutral wire. To solve it in Matlab, it is necessary to

express them in the following way:A scheme of an induction motor with ∆ connected

windings is given in Figure 4.

( )

⋅

=′

t d dv

dt dv

dt di

dt di

dt di

dt di

m

D D D D D

C C C C C

B B B B B

A A A A A

t b

a

B

A

22

54321

54321

54321

54321

/

/

/

/

/

/

00000

00000

0

0

0

0

π

τ

xM (13)

where the following notations are used:

( ) ( S BS A M L BM L A −=−= 2;2 21 )

)

;

( ) ( Rb Ra M L DM LC −=−= 2;2 43 ; Figure 4. Scheme of an induction motor with ∆ connected

windings;; 12 S AS B M L BM L A −=−=

39




Equations of voltage equilibrium for this circuit are: User’s graphical interface window is shown inFigure 4. All parameters of winding: resistance, induc-tance, supply voltage are to be set for each windingseparately. It is required to the type of a motor (rotating or linear) and connection of stator (inductor)windings.

+Ψ=

+Ψ

=

+Ψ

=

.C C C

CA

B B B

BC

A A A

AB

i Rdt

d u

i Rdt

d u

i Rdt

d u

(14)

Figure 5 presents a force developed by LIM atstarting when windings are Y connected without

neutral wire.All other equations such as rotor voltage equilib-rium, magnetic flux linkages for stator and rotor arethe same, as for Y connection with neutral wire.

Therefore, a motor is modelled according to the sameequations, only the right-hand side vector is expressed

as:

( )

−

−

−

−

−

−

−

=

st em

cc

bb

aa

C C CA

A B BC

A A AB

F F

v

i R

i R

i R

i RU

i RU

i RU

xt ,F . (15)

Figure 5. Developed electromagnetic force at starting process

The program gives a possibility to obtain other characteristics, such as speed, inductor and secondarycurrents, excursion and so on. Currents of secondary

at starting no-load LIM are presented in Figure 6.5. Modelling of a linear induction drive

Equations describing behaviour of a motor are programmed in Matlab program language. The deve-loped program is generalized and fits to model opera-tion of induction motors with different connection of

windings. The program can be controlled by Com-mand window row or by user’s graphical interface. It

is recommended to use the command line to investi-gate complex operation modes as reverse or to attachthe developed motor model to other programs. User’sgraphical interface is applied to investigate the simp-lest operating modes as starting, steady state becauseusers freedom in graphical interface is limited.

Figure 6. Transient currents of secondary at starting of no-load linear induction motor

Figure 4. User’s graphical interface windowFigure 7. A phase current during starting process at Y

connection of windings

40



Dynamic Models of Linear Induction Drives

All data of simulation are obtained in commandline, therefore they can be easy processed and used in

new calculations.

6. Conclusions

1. The developed model of linear induction motor considers electric and magnetic asymmetry of LIM due to open magnetic circuit of LIM.

2. The developed algorithm and programs give a possibility to investigate transients in the driveswith rotating and translating motors atsymmetrical and asymmetrical supply voltages,different phase angles between them and differentmodes of winding connection.

3. Created program gives a possibility to investigatetransients of three phase linear drive withconsideration of asymmetry of LIM windings.

4. Due to absence transform of coordinates thedeveloped model facilitates investigation of LIM

control modes.

References

[1] P.-K. Budig. The Application of Linear motors. The

third International Power Electronics and motionControl conference 2000, Vol .3, 1336-1341.

[2] I. Boldea. Linear Electromagnetic actuators and their control. Proceedings of 10th International power

electronics and motion control Conference. Cavtat & Dubrovnik , 2002, 12-20.

[3] J.F. Gieras, P.D. Hartzenberg, I.J. Magura, M.Wing. Control of an elevator drive with a single-sided

linear induction motor. Fifth European Conference on

Power Electronics and Applications, 1993, Vol .4,353-358.

[4] A. Poška, R. Rinkevičienė, Č. Teišerskas. Tiesia-eigės asinchroninės pavaros skaitmeniniai dinaminiai

modeliai. Elektronika ir elektrotechnika. ISSN 1392-1215, Kaunas: Technologija, 1997, Nr .1(10), 50-54.

[5] A. Poška, R. Rinkevičienė, Č. Teišerskas. Tiesiaei-

gės elektros pavaros impulsinis valdymas. Elektro-

technika: Mokslo darbai, Kaunas: Technologija, 1997,

t . 21(30), 137-140.

[6] R. Rinkevičienė, S. Lisauskas. Tiesiaeigių mechatro-ninių sistemų modeliai. Elektronika ir elektrotechnika. Kaunas: Technologija. 2003, Nr .4(46), 69–73.

[7] R. Rinkevičienė, A. Petrovas. Tiesiaeigės asinchro-ninės pavaros modelis. Elektronika ir elektrotechnika. Kaunas: Technologija. 2004, Nr .1(50), 28–32.

[8] R. Rinkevičienė, A. Petrovas. New model of linear

induction drive. Elektronika ir elektrotechnika.

Kaunas: Technologija. 2004, Nr .2(51), 25–28.

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modelo dianamico de induccion lineal