Field-oriented control scheme for linear inductionmotor with the end effect

G. Kang and K. Nam

Abstract: The movement of a linear induction motor (LIM) causes eddy currents in the secondaryconductor sheet at the entry and the exit of the primary core. The eddy currents of the sheet tend toresist sudden ux variation, allowing only gradual change along the airgap. Hence, the so-calledend effect causes not only the losses but also airgap ux prole variation changes depending onthe speed. In this work, an equivalent circuit model of LIM is developed following Duncans perphase model. It is then transformed into a synchronous reference frame which is aligned with thesecondary ux. Also, a eld orientation control scheme is developed which accounts for the endeffect. The validity of the proposed LIM model is demonstrated by comparing experimental andsimulated voltage current relations. With the proposed control scheme, the (secondary) ux-attenuation problem due to the end effect is shown to be resolved in the high-speed range.

1 Introduction

In a linear induction motor (LIM), the primary windingcorresponds to the stator winding of a rotary inductionmotor (RIM), while the secondary corresponds to the rotor.Normally, the secondary part of a LIM is made ofaluminum and steel plates. As the primary moves, a newux is continuously developed at the entry of the primaryyoke, while existing ux disappears at the exit side. Suddengeneration and disappearance of the penetrating eld causeseddy currents in the secondary plate. This eddy currentaffects the airgap ux prole in the longitudinal direction.The loss, as well as the ux-prole attenuation, becomeseverer as the speed increases. Such a phenomena is calledend effect of LIM.Many researchers have derived per-phase equivalent

circuits reecting the end effect [111]. Field theory wasutilised in developing the lumed parameter LIM model[14], in which end effect, eld diffusion in the secondarysheet, skin effect and back-iron saturation were considered.However, the resulting lumped-parameter models look verycomplicated for practical use of modelling and control.Gieras et al. [5] and Faiz et al. [6] developed an equivalent

circuit by superposing the synchronous wave and thepulsating wave caused by the end effect. Nondahl et al. [7]derived an equivalent-circuit model from the pole-by-polemethod based on winding functions of the primarywindings. Several different models were developed fromthe electromagnetic relation in the airgap through aFourier-series approach [810].

Duncan [11] developed the per-phase equivalent circuitby modifying the magnetising branch of the equivalentcircuit of the RIM. Noting that the airgap ux grows anddecays gradually from the entry and the exit, respectively,he introduced a weighting function with the use of adimensionless factor Q. Then, the average values were takento describe the magnetising inductance and the resistance ofthe eddy loss. In this paper, we transform the Duncansmodel into the dq model in a synchronous frame based onthe secondary ux and derive a vector control scheme.

2 End effect of linear induction motor

Figure 1a shows a view of a LIM. Both generation anddecay of the elds cause the eddy current in the secondarysheet. The eddy current in the entry grows very rapidly tomirror the magnetising current, nullifying the airgap ux atthe entry. On the other hand, the eddy current at the exitgenerates a kind of wake eld, dragging the moving motionof the primary core. The density prole of the eddy currentalong the length of LIM looks like Fig. 1c [11]. Hence, theresulting magneto motive force (MMF), and thereby airgapux, looks like Fig. 1b. Duncan introduced a dimensionlessfactor Q, which is dened by

Q9TvTy

l=vyLy=Ry

1

where l, vy, Ly and Ry are the motor length, velocity,secondary self inductance and secondary resistance, respec-tively [11]. Tv is the time for the primary core to traverse apoint in the rail and TyLy/Ry is the time constant of thesecondary circuit. Q represents the motor length for a givenspeed vy, but normalised by the secondary time constant.On this basis, the motor length is innite at zero speed, andwill shrink as the speed increases. Note that if we scale theposition x0 by

x x0=vy

Ly=Ry

the motor airgap resides on the interval, [0, Q]. On thisinterval, the amplitudes (envelope) of eddy and magnetising

G. Kang is with the Electric Powertrain System Development Team, AdvancedR&D Center, Hyundai Motor Company, 772-1, Changduk-Dong, WhasungCity, Kyunggi-Do, 445-850, South Korea

K. Nam is with the Department of Electrical Engineering, Pohang University ofScience and Technology, San-31, Hyoja-dong, Pohang, Kyoungbuk, 790-784,South Korea

E-mail: gubae@hyundai-motor.com

r IEE, 2005

IEE Proceedings online no. 20045185

doi:10.1049/ip-epa:20045185

Paper rst received 14th October 2004 and in nal revised form 10th March2005

IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005 1565

currents were assumed to be given by Imex; Im1 ex,respectively. Based on this proposition, Duncan obtained

the average values for the eddy current, the magnetisingcurrent and secondary eddy loss, and derived a per-phasemodel [11].In Duncans model, the series resistance reecting the

eddy loss appears in the series with the magnetisinginductance. The eddy current loss at the entry andexit is, in principle, the same as the core loss. The coreloss is normally represented by a parallel resistor tothe magnetising inductor [12]. Therefore, the eddycurrent loss of the secondary conductor also needs to bedescribed by a parallel resistor. However, in Duncansmodel, the eddy current loss is described by a seriesresistor.As an equivalent-circuit model for LIM with the end

effect, consider Fig. 2. Following Duncans model, let themagnetising inductance be equal to Lm0(1f(Q)), where Lm0is the magnetising inductance at zero speed andf(Q) (1eQ)/Q.We neglect the eddy-current loss Reddy for the sake of

simplicity. However, the inclusion of the loss branchincreases the complexity of the equation greatly, whilstthe presence or absence of the loss branch does notmake a signicant difference in the current dynamics unlessLIM is moving at a very high speed. Note that onlyDuncans magnetising inductance model is utilised here,whilst the resistance Reddy reecting eddy current loss isneglected.

entry eddy current

primary

secondary back iron

normalised time

airg

ap fl

uxe

ddy

curre

ntby

the

end

effe

ct

normalised time

secondary aluminum plate

exit eddy current

0

0

Q

Q

X

X

ex

1ex

vy

a

b

c

Fig. 1 Linear induction motora Eddy-current generation at the entry and exit of the airgap when theprimary moves with speed nyb Airgap ux prolec Eddy-current density porle along the length of LIM [11]

Rx

iax

ibx

ibx

iay

iby

icy icy

iby

iay

vax

vbx

Llx

Rx Llx

Rx Llx Lly

Lly

Lly

Ry

Ry

Ry

Reddy

Reddy

Reddy

Lm0{1-(Q)}

Lm0{1-(Q)}

Lm0{1- (Q )}

0.5Lm0{1-(Q)}

0.5Lm0{1- (Q)}

0.5Lm01-(Q)}

+

+

vcx

+

ideal transformer

ideal transformer

ideal transformer

Fig. 2 Per-phase equivalent circuit of LIM considering the end effect

1566 IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005

The voltage equations of the per-phase model shown inFig. 2 are given by

vabcx Rxiabcx pkabcx 2v0abcy Ry i0abcy pk0abcy 3

where vabcx vax vbx vcxT ; iabcx iax ibx icxT andkabcx [lax lbx lcx]T are the voltage, current and uxlinkage of the primary, and v0abcy v0ay v0by v0cy T ; i0abcy i0ay i0by i0cy T and k0abcy l0ay l0by l0cy T are the voltage,current and ux of the secondary in the secondary frame.Rx denotes the primary resistance. Further, p denotes thedifferential operator, d/dt.To keep the same convention as for the rotary motor, it is

assumed that the primary is xed, while the secondary ismoving. The ux-linkage equations for the primary andsecondary are given by

kabcx Llx Lmxiabcx Lmy i0abcy 4

k0abcy Lly Lmxi0abcy LmyT i0abcx 5where

Llxly Llxly 0 00 Llxly 00 0 Llxly

264

375

Lmx Lm01 f Q1 1

2 1

2

12

1 12

12

12

1

264

375

Lmy Lm01 f Qcos yy cos yy 2p3

cos yy 2p3

cos yy 2p3

cos yy cos yy 2p3

cos yy 2p3

cos yy 2p3

cos yy

264

375

Llx(ly) denotes the primary (secondary) leakage inductanceand yy is the angular displacement of the secondary.Co-ordinates are changed into a synchronous reference

frame with the following transformation map:

f edxf eqx

" # 23

cos oet sin oet

sin oet cos oet

1 1

2 1

2

03

p2

3

p2

" # faxfbxfcx

264

375

6

where oe is the electrical angular velocity and f is v, i or l.Utilising the transformation map (6), we obtain from (2)and (3) such that

vedx Rxiedx pledx oeleqx 7

veqx Rxieqx pleqx oeledx 8

vedy Ryiedy pledy oe oyleqy 9

veqy Ryieqy pleqy oe oyledy 10where vedx; veqx; iedx; ieqx and ledx; leqx are the dq-axisvoltages, currents and ux linkages of the primary, andvedy ; veqy; iedy ; ieqy and ledy ; leqy are the dq-axis voltages,currents and uxes of the secondary in the secondary frame.oy denotes the secondary angular speed. The superscript eon v, i, or l denotes the quantity in the synchronous

reference frame. The dq-axis ux linkages of the primaryand secondary are given by

ledx Llxiedx Lm 1 f Qf giedx iedy 11

leqx Llxieqx Lm 1 f Qf gieqx ieqy 12

ledy Llyiedy Lm 1 f Qf giedx iedy 13

leqy Llyieqy Lm 1 f Qf gieqx ieqy 14where Lm 1.5Lm0.We further assume that the reference frame is aligned to

the secondary ux, i.e. we use the secondary ux-orientation

scheme. Then, leqy _leqy 0. With vedy veqy 0, weobtain from (7)(10) such that

piedx Rx

LsQ iedx

1

LsQ vedx oeieqx 15

pieqx Rx

LsQ ieqx

1

LsQ veqx

oe iedx Lm1 f Q

LsQ Ly Lmf Q ledy

" #16

ledy Lm 1 f Qf g

1 Ty Lmf Q=Ry

piedx 17

osl Lm 1 f Qf gieqx

Ty Lmf Q=Ry

ledy18

where the total leakage inductance is denoted by

LsQ Lx Lmf Q Lm 1 f Qf g 2

Ly Lmf Qand osloeoy, the slip angular frequency. Note that ifvy 0 with f Q 0, then osl Lmieqx=Tyledy and ledy Lmiedx=1 Typ which are the cases of RIM. Note from(17) that ledy for a given i

edx decreases as LIM velocity vy

increases since f(Q)-1 as vy-N. This decrease of themagnetising inductance is an illustration of the airgap uxattenuation due to the end effect. Hence, to keep the uxconstant, iedx needs to be compensated as motor speedincreases.On the other hand, input power Pin is obtained from

(7)(10) such that

Pin 32

vedxiedx veqxieqx vedyiedy veqyieqy

32

Rx iedx

2 ieqx2

Ry iedy2 ieqy2

:

p Llx2

iedx2 ieqx2

Lly

2iedy

2 ieqy2

Lm 1 f Qf g iedx iedy 2

ieqx ieqy 2

orLm 1 f Qf g ieqxiedy iedxieqy h i

19Note that the electrical power into the terminals of LIMcan be divided into three groups. The rst group accounts

IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005 1567

for the power dissipated in the primary and secondaryresistances. The second group corresponds to the time rateof the magnetic energy stored in the inductances. The lastgroup Pem 1:5orLmf1 f Qg ieqxiedy iedxieqy is theelectromechanical output power being converted fromelectrical to mechanical form. Since Pem is equal to theproduct of speed and torque, thrust F is obtained from (13)and (14) such that

F ptPemom

32

P2

ptLm 1 f Qf gLy Lmf Q l

edyi

eqx 20

where t denotes the pole pitch and om the mechanicalangular speed, i.e. om (2/P)or. In deriving the abovethrust equation, it was assumed that leqy 0 with thesecondary ux-orientation scheme.Since Ly Lm Lly ; Ly Lm if the leakage inductance

Lly{Lm. In this case 1f(Q) cancels out, and thereby Fdoes not directly depend on Q. Therefore, the dependenceof F on the speed appears not to be so signicant as far asledy is kept constant. It will be shown in the later simulationpart that dependence of ledy on Q is large, while dependenceof F on Q is small.

3 Field-orientation control based on secondaryflux

In Section 2, we developed a dq dynamic model for LIMwith the end effect. In this Section, we construct a speed andthrust controller based on the proposed model.

3.1 Flux and thrust controllersWith the classical vector control, secondary ux is linearlyproportional to primary d-axis current in the steady state,i.e. ledy Lmiedx. However, with the proposed model thecoefcient depends strongly on the velocity, i.e. it followsfrom (17) that ledy Lm1 f Qiedx in the steady state.Although the coefcient varies with velocity, it still holdslinear proportionality, and a proportionalintegral (PI)controller performs well for this case unless the rate ofchange of Q is fast. However, the point is how to estimatethe ux for the feedback. Utilising (17), we obtain a ux

controller such that

iedx PI1 ledy l^edy

PI1 ledy Lm 1 f Qf g

1 Ty Lmf Q=Ry

piedx

" # 21

where PI1 denotes a PI controller of the type (KP+KI/p)for some KP, KI40, iedx is a d-axis primary currentcommand, ledy is a d-axis secondary ux command, andl^edy is an estimate of l

edy . The ux estimate is equivalently

expressed as

pl^edy Ry

Ly Lmf Q l^edy

RyLm 1 f Qf gLy Lmf Q i

edx: 22

Similarly to the ordinary vector-control case, a thrustcontroller is developed with a PI controller such that

ieqx PI2F F^

PI2 F 32

P2

ptLm 1 f Qf gLy Lmf Q l^

edyi

eqx

23

where PI2 denotes another PI controller, ieqx is a q-axisprimary current command, F* is a thrust command, and F^is an estimate of the thrust.

3.2 Current controllerThe primary currents are regulated by a PI controller (PI3)with decoupling and back-EMF compensation terms suchthat

vedx PI3 iedx iedx oeLsQieqx 24

veqx PI3 ieqx ieqx

oeLsQiedx

oe Lm 1 f Qf gLy Lmf Q l^edy : 25

The overall proposed control block diagram and LIMmodel appear in Fig. 3. Recall that the end effect is countedin both the LIM model and the controller through Q.

LIM model with end effect

3p4

Lm{1(Q)}Ly Lm(Q)

dy

3p4

Lm{1(Q)}Ly Lm(Q)

dy

Lm{1(Q )}Ly Lm(Q)

dy (Q)e

dy idx idx

idx

vdx

vqx

iqx

iqxiq

dydy

F

F

Lm{1-(Q)}1+TyLm(Q)/Ry)p

Lm{1-(Q)}1+Ty Lm(Q)/Ry)p

Lm{1(Q)}Ly Lm(Q )

dy e

eL(Q) eL(Q)

eL(Q) eL(Q)

PI1

PI2

PI3

PI3

current controller

Rx + L(Q)P1

Rx + L(Q )P1

+

+

+

+

++

+

+

+

+

+

Q

F

Q

e* e*

e*

e

e

e

e

e

e

e

e

e

e

e

e

Fig. 3 Proposed controller and LIM model considering the end effect

1568 IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005

4 Simulation and experimental results

4.1 Simulation resultsFor the simulation study, we use (2)(5) in the abc frameas a LIM model. Simulation was performed with themeasured parameters of a real LIM that were utilised in theexperiment. The measured parameters were: Llx 22.5mH,Lly 6.5mH, Lm 37.6mH, Rx 1.2O, Ry 2.7O andmotor power 4.4kVA.Figure 4 shows the magnetising inductance Lm{1f(Q)}

and thrust coefcient Lm{1f(Q)}/{LyLmf(Q)} againstspeed. A noticeable reduction of the magnetising inductanceis observed with increase of vy, whereas the thrust coefcientremains almost invariant. This supports the observationmentioned in the last part of Section 2. Specically, themagnetising inductance reduces to about a half of thestarting value at a speed of 10m/s.Figure 5 presents simulation results showing the

responses to a rectangular shape-thrust command. Figure5a shows the plots of thrust, primary currents, secondaryux and speed when the conventional vector control isapplied, whereas Fig. 5b shows the same things whenthe proposed control is applied. The term conventionalvector control implies the eld-oriented control withouta provision for the end effect. With the conventionalvector control, ux level decreases as the speed increases.Correspondingly, the thrust also decreases, yieldinglower speed. On the other hand, with the proposedscheme the ux level is kept constant and no discrepancyis observed between the thrust command and theproduced thrust. Note however, that the d-axis-currentcommand level increases with the speed to compensatefor the end effect. The proposed scheme produces a highernal speed.

Lm{1-(Q )}

0 101.886 0.943 0.623

20 30

vy , m/s

vy , m/s

(Q)

0 101.886 0.943 0.623

20 30 (Q)

1

0

thru

st c

oeffi

cient

40

0

ma

gnet

ising

indu

ctan

ce, m

H

Lm{1-(Q )}

LyLm(Q ))

a

b

Fig. 4 Magnetising inductance and thrust coefficienta Magnetising inductance against speedb Thrust coefcient against speed

800

0200

F F F

0 100time, s

0 100time, s

0 100

0 100

time, s

time, s

800

0

2000 100time, s

0 100time, s

0 100time, s

0 100

4.5 m/s

time, s

iqx iqx

dydy

vy vy

idx idx

20

05

20

05

0.25

0

0.25

0

thru

st, N

mcu

rrent,

Aflu

x, W

bsp

eed,

m/s

thru

st, N

mcu

rrent,

Aflu

x, W

bsp

eed,

m/s

8

02

8

02

a b

F

3.9 m/s

e e

e

e

e

e

Fig. 5 Simulation results when a square shape-thrust command is appliedF thrust, iedx; iedy primary currents, ledy secondary current, ny speeda Conventional controlb Proposed control

IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005 1569

4.2 Experimental resultsThe linear motor used in this experiment is shown in Fig. 6.The primary yoke was mounted on a xed pole, while thesecondary circuit comprising aluminum plate and steel wasmounted on a big ywheel. The dimensions of the LIM areshown in Table 1. Figure 7 shows a block diagramillustrating the proposed vector control. In Fig. 7, PIgains for the ux, thrust and current controllers wereselected as PI1 (200p+100000)/p, PI2 (0.015p+15)/p,and PI3 (25p 80000)/p, respectively.

The validity of the proposed model is demonstrated rstby comparing the measured current and the calculatedcurrents based on two models, while increasing the speed.Figure 8 shows the currents in the steady state at each speedfor the same voltage inputs: Fig. 8 shows real measured

currents iedx ; ieqx, the calculated currents i^

edx; i^

eqx based on the

proposed model (15)(18), and the calculated iedx;ieqx based

on the conventional model without the end effect. Note thatthe proposed model produced results very close to theexperimental results, while the conventional model showedbig difference as the speed increased.Figure 9 shows the experimental results when the same

rectangular shape-thrust command was applied. Theexperimental results look quite close to the simulationresults shown in Fig. 5. Note again that the conventionalcontrol results in a decrease in ux as the speed increases,but the proposed control yields constant ux by injectingmore iedx.

Fig. 6 LIM with a disk-type secondary plate used in thisexperiment

vdxe

iqx

iqx

iqx

iqxe

idxe

iqx

iqxe

vqxe

vy

/

vcx

vDC

iax

e

ysl

e

e

ibx

vbx

vax

dy

dy

dy

F

F

F +

+ +

++

3PLm{1(Q)}4{Ly Lm(Q)} dy

iqx

Lm{1(Q)}idx1+{Ty Lm(Q)/Ry}P

Lm(1(Q))Ty Lm(Q)/Ry

iqxdy

PI2PI

PI2

++

vy

vy

field weakeningprofile

estimates

LIMdq

abc

dq

abc

curr

ent c

ontro

ller w

ith d

ecou

plin

gand

back

EM

F co

mpe

nsat

ion

consi

derin

g en

d ef

fect

invertorabc/dq transformationspeed controller

TMS320C32 DSP board

e* e*

e*

ee

e e

e

e

e

e

e

Fig. 7 Block diagram of the proposed vector-control scheme considering the end effect

Table 1: Dimensions of a single-side LIM

Motor length, width 0.308m,0.170m

Series turns/phase 240

Pole pitch 0.066m Primary back iron 15mm

Number of poles 4 Effective airgap 6.6mm

Slots per pole 6 Secondary sheet 3mmaluminum

Coil pitch 5 Q 18.86/vy

15

5

0

5

10

prim

ary

curre

nts,

A

speed, m/s1.60 3.2 4.8 6.4 8.0

idx , iqxidx , iqx

: proposed model

idx , iqx : conventional model

e e

e

e

e

e

Fig. 8 Experimental results

Measured currents iedx; ieqx, calculated currents i^

edx; i^

eqx based on the

proposed model, and calculated currents iedx; ieqx based on the

conventional model without the end effect. All steady-state valuesfor the same voltage inputs at each speed.

1570 IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005

Figure 10 shows the experimental results when a staircasespeed command, [0, 1.6, 3.2, 4.8]m/s was applied. At thistime LIM was running in a speed-control mode. Shown inthe last row are the acceleration periods and the duration ofinjecting the maximum ieqx. With the conventional con-troller, the period gets longer and longer as the speedincreases. The reason is that the ux level, and consequentlythe thrust, decreases as the speed increases. In Fig. 10b theacceleration period is almost identical irrespective of speedwith the proposed control method. This indirectly provesthe effectiveness of the proposed control scheme thatdecouples ux level from speed, i.e. the ux level is madeindependent of the speed.

5 Concluding remarks

The end effect of LIM causes prole change andattenuation of the airgap ux. This becomes more severeas the speed increases. In this work Duncans model is

modied into the d-q frame and the eld-oriented controlscheme aligned with the secondary ux was developed.Both ux and thrust coefcient were expressed with Q.However, it was observed in the LIM model that thedependence of the ux on Q was great, whereas thedependence of the thrust coefcient on Q was not sosignicant. The validity of the LIM model was checkedby comparing currents from an experiment with those fromthe models. A vector controller was developed based on theproposed model. Experiments were performed in the thrust-control mode, as well as in the speed control mode. Inboth cases, the problem of ux attenuation caused by speed(eddy current) increase was not observed with the proposedcontrol scheme.

6 References

1 Dawson, G.E., Eastham, A.R., and Gieras, J.F.: Design of linearinduction drives by elds analysis and nite-element techniques,IEEE. Trans., 1986, IA-22, (5), pp. 865873

i eqx ieqx

i edx i edx

dyedy

vy vy

0

0

0

0

0

00

[5A/division] [5A/division]

[0.05Wb/division] [0.05Wb/division]

[2m/s/division] [2m/s/division]

time, s100

a

0time, s

100

b

e

Fig. 9 Experimental resultsPrimary currents iedx; i

eqx, secondary ux l

edy , and motor speed ny when a rectangular shape thrust command is applied

a Conventional controlb Proposed control

vy* vy*

vy

idx idx

iqxiqx

vy

(2m/s/division) (2m/s/division)

(2m/s/division)(2m/s/division)

(2.5A/division) (2.5A/division)

(5A/division) (5A/division)

30.4s 37.8s 43.3s 30.4s 30.6s 30.8s

0

0

0

0

0

0

0

0

0 200time, s

0 200time, s

a b

e

ee

e

Fig. 10 Experimental resultsMotor-speed command vy , motor speed ny, and primary currents i

edx; i

eqx when a staircase speed command, [0, 1.6, 3.2, 4.8]m/s is applied

a Conventional controlb Proposed control

IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005 1571

2 Pai, R.M., Boldea, I., and Nasar, S.A.: A complete equivalent circuitof a linear induction motor with sheet secondary, IEEE. Trans., 1988,MAG-24, (1), pp. 639654

3 Gerias, J.F., Eastham, A.R., and Dawson, G.E.: Performancecalculation for single-sided linear induction motors with solid steelreaction plate under constant current excitation, IEEE. Proc. Electr.Power Appl., 1985, 132, (4), pp. 185194

4 Idir, K., Dawson, G.E., and Eastham, A.R.: Modeling andperformance of linear induction motor with saturable primary, IEEE.Trans., 1993, IA-29, (6), pp. 11231128

5 Gieras, J.F., Dawson, G.E., and Eastham, A.R.: A new longitudinalend effect factor for linear induction motors, IEEE. Trans., 1987,EC-22, (1), pp. 152159

6 Faiz, J., and Jafari, H.: Accurate Modeling of single-sided linearinduction motor considers end effect and equivalent thickness, IEEE.Trans., 2000, MAG-36, (5), pp. 37853790

7 Nondahl, T.A., and Novotny, D.W.: Three-phase pole-by-pole modelof a linear induction machine, IEE Proc. Elect. Power Appl., 1980,127, (2), pp. 6882

8 Yoshida, K. et al.: New transfer-matrix theory of linear inductionmachines, taking into account longitudinal and transverse ferromagneticend effects, IEE Proc. Elect. Power Appl., 1983, 128, (5), pp. 225236

9 Balchin, M.J., and Eastham, J.F.: Model for transients in linearinduction machines, IEEE. Trans., 1997,MAG-33, (5), pp. 41914193

10 Mori, Y., Torii, S., and Ebihara, D.: End effect analysis of linearinduction motor based on the wavelet transform technique, IEEE.Trans., 1997, MAG-33, (5), pp. 41914193

11 Duncan, J., and Eng, C.: Linear induction motor-equivalent circuitmodel, IEE Proc., Electr. Power Appl., 1983, 130, (1), pp. 5157

12 Levi, E., Boglietti, A., and Pastrorelli, M.: Iron loss in rotor-ux-oriented induction machine: Identication, assessment of detuning,and compensation, IEEE. Trans., 2002, PE-38, (2), pp. 13651370

1572 IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005